Aspects of AdS / CFT - Elektronische Dissertationen der LMU

Transcription

Aspects of AdS/CFT:
Black Solutions in Gauged Supergravity
and Holographic Conductivities
Susanne Barisch-Dick
München 2013
Aspects of AdS/CFT:
Black Solutions in Gauged Supergravity
and Holographic Conductivities
Dissertation
an der Fakultät für Physik
der Ludwig–Maximilians–Universität
München
vorgelegt von
aus Regensburg
München, den 18.02.2013
Erstgutachter: Prof. Dr. Dieter Lüst
Zweitgutachter: Dr. Michael Haack
Tag der mündlichen Prüfung: 26.04.2013
Contents
Zusammenfassung
vii
1 Introduction and Overview
1
2 Black Holes and Attractors
2.1 The attractor mechanism . . . . . . . . . . . . . . . . . . . . .
2.2 Black holes as BPS states . . . . . . . . . . . . . . . . . . . .
2.3 Special geometry . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 The scalar geometry for N = 2 supergravity in five dimensions
2.5 Gauged supergravity in four and five dimensions . . . . . . . .
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3 The
3.1
3.2
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3.5
3.6
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AdS/CFT Correspondence
The field theory side of the duality . . . . . .
The gravitational side of the duality . . . . . .
The AdS/CFT correspondence . . . . . . . . .
The AdS/CFT dictionary . . . . . . . . . . .
Generalizations and applications of AdS/CFT
Linear response . . . . . . . . . . . . . . . . .
Basic aspects of holographic superconductivity
4 Holographic Flavor Conductivity
4.1 Setup . . . . . . . . . . . . . . .
4.2 Conductivities . . . . . . . . . .
4.2.1 Analytical solution . . .
4.2.2 Numerics . . . . . . . .
4.3 Discussion of our results . . . .
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5 Nernst branes in four-dimensional gauged supergravity
5.1 Flow equations for extremal black branes in four dimensions
5.1.1 First-order flow equations in big moduli space . . . .
5.1.2 Generalizations of the first-order rewriting . . . . . .
5.1.3 AdS2 × R2 backgrounds . . . . . . . . . . . . . . . .
5.2 Exact solutions . . . . . . . . . . . . . . . . . . . . . . . . .
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5.3
5.2.1 Solutions with constant γ . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Solutions with non-constant γ . . . . . . . . . . . . . . . . . . . . .
Nernst brane solutions in the STU model . . . . . . . . . . . . . . . . . .
6 Extremal black brane solutions in gauged supergravity in D = 5
6.1 First-order equations for stationary solutions in five dimensions . .
6.1.1 Flow equations in big moduli space . . . . . . . . . . . . . .
6.1.2 Hamiltonian constraint . . . . . . . . . . . . . . . . . . . . .
6.2 Reducing to four dimensions . . . . . . . . . . . . . . . . . . . . .
6.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Solutions with constant scalar fields X A . . . . . . . . . . .
6.3.2 Solutions with non-constant scalar fields X A . . . . . . . . .
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7 Conclusions and Outlook
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A Special geometry
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B First-order rewriting in minimal gauged supergravity in D = 4
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C Einstein equations in five dimensions
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D Relating five- and four-dimensional flow equations
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E A different first-order rewriting in five dimensions
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Danksagung
162
Zusammenfassung
Der Themenbereich dieser Dissertation gliedert sich in zwei Teilgebiete, die beide durch
die AdS/CFT-Korrespondenz motiviert sind. Zum einen werden neue schwarze Gravitationslösungen im Bereich geeichter Supergravitation in vier und fünf Raumzeitdimensionen
konstruiert. Zum anderen werden Methoden aus der Korrespondenz zwischen Eich- und
Gravitationstheorien verwendet um holographisch den frequenzabhängigen Leitfähigkeitstensor für bestimmte Feldtheorien zu berechnen.
Den Ausgangspunkt zur Berechnung dieses Leitfähigkeitstensors bildet eine SchwarzeLoch-Lösung der vierdimensionalen Einstein-Yang-Mills-Wirkung mit SU (2)-Eichfeld. Die
SU (2)-Symmetrie wird durch die Wahl des Eichfeldes explizit gebrochen und auf eine U (1)Symmetrie in τ 3 -Richtung reduziert. Es wurde bereits in anderen Arbeiten nachgewiesen,
dass dieses System bei einer kritischen Temperatur einen Phasenübergang durchläuft, in
dessen Folge die verbliebene U (1)-Symmetrie spontan gebrochen wird und ein schwarzes
Loch mit Vektor-Haar entsteht. Dieses Ergebnis wurde dahingehend interpretiert, dass in
der dualen Feldtheorie ein Phasenübergang zu einem p-Wellen-Supraleiter stattfindet.
Diese Doktorarbeit befasst sich mit den Strömen in τ 1 - und τ 2 -Richtung in der normalen, d.h. nicht-supraleitenden, Phase. Der frequenzabhängige Leitfähigkeitstensor wird
mit Hilfe von Linearer-Response-Theorie basierend auf holographischen Methoden berechnet, die im Rahmen der AdS/CFT-Korrespondenz entwickelt wurden. Dabei konnte festgestellt werden, dass innerhalb eines bestimmten Frequenzbereichs ein negativer Eigenwert
des hermiteschen Anteils des Leitfähigkeitstensors auftritt, was zu negativer Entropieproduktion zu führen scheint. Zum jetzigen Zeitpunkt konnte dieser Effekt noch nicht hinreichend geklärt werden.
Der zweite Teil dieser Dissertation hat die Suche nach schwarzen Lösungen mit AdSAsymptotik innerhalb von N = 2 U (1)-geeichter Supergravitation in vier und fünf Dimensionen zum Inhalt. Hierbei werden nur Vektor-Multipletts berücksichtigt. Derartige
Lösungen sind von Interesse, da schwarze Geometrien mit AdS-Asymptotik im Rahmen
der Korrespondenz zwischen Gravitation und Eichtheorien Feldtheorien bei endlicher Temperatur und Ladungsdichte beschreiben.
Sowohl in vier als auch in fünf Dimensionen wird auf den Attraktor-Mechanismus
für extremale schwarze Geometrien zurückgegriffen. Dieser garantiert, dass die relevanten Bewegungsgleichungen gewöhnliche Differentialgleichungen erster Ordnung sind. Diese
viii
Flussgleichungen stellen eine Verallgemeinerung des asymptotisch flachen Falls dar auf den
Fall mit AdS-Asymptotik. Im Folgenden werden unterschiedliche neue schwarze Lösungen
konstruiert. In beiden Dimensionen konnten extremale schwarze Branen mit verschwindender Entropiedichte gefunden werden. Diese werden in Anlehnung an das Nernst-Gesetz
der Thermodynamik als Nernst-Branen bezeichnet. Extremale schwarze Lösungen haben
die Eigenschaft, dass ihre Temperatur gleich null ist. Viele bekannte extremale schwarze
Lösungen wie Reissner-Nordström-Lösungen weisen jedoch einen nicht-verschwindende Entropie auf. Daher sind die Nernst-Lösungen von Interesse, die bei verschwindender Temperatur eine verschwindende Entropiedichte besitzen.
Weiterhin werden Konfigurationen untersucht, die keinen Nernst-Lösungen entsprechen.
Hier konnten in vier Dimensionen sowohl singuläre Lösungen gefunden werden als auch eine
numerische Lösung, die zwischen AdS2 × R2 und AdS4 interpoliert. In fünf Dimensionen
werden analytische Lösungen mit konstanten Skalaren sowie numerische Lösungen mit
nicht-konstanten Skalaren konstruiert. Diese Lösungen verhalten sich asymptotisch wie
AdS5 und können am Horizont durch eine BT Z × R2 -Geometrie beschrieben werden.
Ferner wird der Zusammenhang zwischen den vier- und fünfdimensionalen Bewegungsgleichungen dargestellt.
Chapter 1
Introduction and Overview
“The most beautiful thing we can experience is the mysterious. It is the source of all true
art and science.”
Albert Einstein, “How I see the world”
With his words Einstein shows the idea of mankind being driven by an intrinsic urge for
knowledge. A similar picture was drawn 150 years before by Wilhelm von Humboldt when
he speaks of what he calls a man’s “Bildungsbedürftigkeit” which may be translated as
the “longing for education”.
For Humboldt education implies a person’s involvement with all the surrounding world,
in the German original called “Aneignung von Welt”. Surely from today’s perspective
exploring the laws of nature is closely related to this idea of education. People feel the
need to know why apples fall down from trees or why there are myriads of stars in the sky.
Another consiredable philosopher, Immanuel Kant, sees man as an entity endowed
with reason. From Kant’s perspective it is possible to gain insight into the “things in
themselves”, which include the natural phenomena, by the use of reason. Of course, we
all differ in our perceptions and processing of information but according to Kant this does
not mean that cognition independently of an observer is impossible. For Kant this is
true especially in mathematics and physics. Here his claim “sapere aude!” – “dare to
know!” – seems to be fulfilled best. By the use of reason we are able to decide whether
a theory describes well the physical phenomena or whether we should discard it. Of
course, this process of acquiring knowledge goes hand in hand with experiences we make
by perception, which from a physicist’s point of view are experiments. Both traits, the
thurst for knowledge and the use of reason, make a scientist.
Within the last 300 years a lot of progress has been made in understanding the laws of
nature. Very often this process was accompanied by a change of paradigm. Some of the
real revolutions in physics have been made because somebody went beyond the boundaries
of existing knowledge – somebody “dared to know”.
Some of the major changes in physics happened at the beginning of the twentieth
century. At that time two theories involving completely new physics were found: Quantum
mechanics and Einstein’s theory of relativity.
2
1. Introduction and Overview
Before it was believed that everything is completely determined by Newton’s laws of
mechanics, statistical mechanics and Maxwell’s theory of electromagnetism. All these
branches had in common the deterministic way of thinking. Pierre-Simon Laplace stated
that an observer who knows the position and velocity of all masses at a given time could
predict everything that will happen in the future. Such an observer who is able to determine
all these quantities at a given moment is known as Laplace’s demon. From this mechanistic
perspective one can see two things which are typical for the leading paradigm in physics
before the beginning of the twentieth century: The movement of any mass or particle
is completely determined by the initial conditions and time as well as space are only
background parameters that are the same for any observer.
Both these ideas were completely overthrown in the following. Einstein’s theory of
relativity states that space and time are different for different observers. There is no
absolute knowledge about these quantities. Furthermore, mass and energy are equivalent
and both backreact on space and time: Masses curve the space around them and influence
in that way the movement of other masses or even light rays. The theory of relativity is
very well tested experimentally, e.g. it was able to explain the mercury precession.
The second fully new theory was quantum mechanics. In contrast to relativity, which
deals with very large structures and distances, quantum mechanics is a theory for microscopically small scales. Quantum mechanics was the end of the deterministic way of
thinking that dominated hitherto. The uncertainty principle states that it is impossible
to measure position and velocity of very small objects at the same time. The more exact
one measures one of these two quantities the more uncertain becomes the other. Laplace’s
demon is in trouble!
Both of these theories – relativity and quantum mechanics – have made an enormous
contribution to understand the laws of nature. But there is one real big problem: A full
description of gravity must necessarily include quantum effects, but up to now there is no
unified description of quantum mechanics and gravity within one theoretical framework.
The whole twentieth century seems at large part devoted to the search for a ”theory of
everything”. Such a theory would unify all known fundamental forces, the electromagnetic
force, the weak and strong force as well as gravity. The principles of quantum mechanics
have successfully been used in the formalism of quantum field theory with which three of the
four fundamental forces can be described. The quantum field theory for the electromagnetic
interaction is QED (quantum electrodynamics). This was unified with the weak interaction, which acts in processes like the nuclear beta decay, in the Glashow-Salam-Weinberg
model. Also the strong force, which describes the interaction of quarks and gluons, can be
formulated as a quantum field theory, namely QCD (quantum chromodynamics).
All our knowledge about the non-gravitational interaction of particles is governed by
the standard model of particle physics. The standard model has been very successful and
it is also very well tested experimentally. All the predicted particles have been found, the
latest discovery being a boson that is very likely to be the Higgs boson in 2012.
But we are still lacking a unified theory for all four forces including gravity. One
approach for such a theory is string theory. In string theory the particles are replaced
by very tiny strings which can vibrate in different modes. The different modes represent
3
different particles. One of those vibrational states is the graviton, the force carrier of
the gravitational field. Gravity is inherently included in string theory and moreover, it
is also a quantum theory. So is string theory ”the theory of everything”? At present we
cannot decide on that. String theory is a very complicated object and up to now there
is no experimental verification of it. It is even hard to obtain an experimentally testable
prediction from string theory. Nevertheless it is a consistent approach for the unification of
all fundamental forces and therefore due to this very deep and fundamental aim it is worth
working on it. Moreover, there is a large interplay between string theory and mathematics
which makes string theory very interesting from a theoretical point of view.
But not only that. In the late 1990ies a duality was found between string theory on
a space called AdS space and field theories living on the boundary of this space. This
conjecture is called the AdS/CFT correspondence. It opened the possibility to apply
string theory, even though in a more indirect way, to strongly coupled field theories. It is
hoped that this approach might lead to new insights into condensed matter systems as for
instance superconductors.
The work presented in this thesis takes the AdS/CFT correspondence as its starting
point. We will especially perform conductivity calculations using AdS/CFT methods and
construct black solutions with AdS asymptotics. The presented results have partly been
published in [1, 2]. To be more precise, this thesis is organized as follows.
Chapters 2 and 3 are introductory chapters. Here the necessary tools which lay the foundation of our work are presented.
This includes basic facts about black holes and branes. Especially the attractor mechanism for extremal black holes is explained. We introduce this mechanism in the context of
N = 2 supergravity. The attractor mechanism forces the values of the fields on the horizon
to be independent of their asymptotic values and fixes them solely in terms of the charges
of the black hole. Furthermore we will give a brief review on the scalar geometry of N = 2
U (1) gauged supergravity in four and five dimensions, namely special geometry and very
special (real) geometry.
In chapter 3 the AdS/CFT correspondence is introduced and an example is given of
how to apply AdS/CFT to compute conductivities in the dual field theory using linear
response. In addition, a short introduction to holographic superconductivity is presented.
Just in the sense of Einstein’s quotation from the beginning we will meet a little bit of art
at some point on our way through these two chapters.
In chapter 4 we compute the frequency dependent conductivity tensor for a theory with
SU (2) flavor symmetry dual to a four-dimensional black hole solution in Einstein-YangMills theory.
Theories with global U (1) symmetry in the field theory have been used in AdS/CFT to
construct toy models for s-wave superconductors holographically. In those approaches the
phase transition to the superconducting state is modeled by the development of a black hole
with scalar hair in the bulk. Hairy black holes are not allowed to exist in asymptotically
flat space due to the no-hair conjecture (cf. e.g. [3]). In spaces which asymptote to AdS,
4
though, it is well possible for a black hole to have hair (cf. e.g. [4]).
Of course, it is interesting to transfer AdS/CFT techniques to theories with non-Abelian
symmetry group as such theories play a vital role in very different fields of physics. For
instance, the original formulation of the AdS/CFT correspondence, which states the equivalence of IIB string theory on AdS5 × S 5 and N = 4 SYM living on the boundary of AdS5
in the large N limit [5], involves a global SU (4) symmetry on the field theory side. This
is the R-symmetry which rotates to SUSY charges into each other.
Furthermore, black hole solutions with AdS asymptotics have been constructed within
four-dimensional Einstein-Yang-Mills theory with SU (2) gauge fields. This setup was used
before in [6] to show that a black hole with vector hair occurs below a certain critical
temperature. In [7] the frequency dependent conductivity for the current in τ 3 direction
in flavor space was computed with AdS/CFT methods. In [7] the authors interpreted the
delta peak at zero frequency as well as the appearing frequency gap as a sign of p-wave
superconductivity.
In contrast to [7] we consider the frequency dependent conductivity tensor in the normal
state. Our calculations involve currents in τ 1 and τ 2 direction in the normal state. We
find that for a certain range of frequencies the Hermitean part of the conductivity develops
a negative eigenvalue and therefore the entropy production rate turns negative. At the
time of writing this thesis we could not find a satisfying explanation of this result. Some
possibilities are discussed at the end of chapter 4.
Also in the following chapters we stay within the AdS/CFT context, but now we concern
ourselves more with the gravitational part of the AdS/CFT correspondence. Our main
field of interest will be to construct black solutions within the framework of N = 2 U (1)
gauged supergravity in four and five dimensions with AdS asymptotics.
Black geometries play an important role in research. In the 1970ies remarkable analogies
between the classical black hole mechanics and the laws of thermodynamics were found.
Because of this, it is possible to assign temperature and entropy to a black hole. For
example, the entropy is proportional to the horizon area of the black hole. We address
this point in chapter 2 in more detail. Some of the most interesting black holes are the
extremal black holes. These have a lot of appealing features. E.g. they obey the attractor
mechanism and display the striking feature of having zero temperature. We will comment
on that more extensively in chapter 2.
Simultaneously, there is another critical way in which black holes play an essential role
in string theory. Within the AdS/CFT framework black holes, which are states in the bulk,
correspond to thermal ensembles in the dual field theory at the same temperature as the
black hole. The dynamics of bulk fields in the black hole background therefore provides
information on the interaction of the corresponding operators in the thermal ensemble of
the dual field theory.
From this point of view it is particularly important to construct black hole solutions in
spaces with AdS asymptotics. Charged black holes with AdS asymptotics were discussed for
instance in [8]. These are the so-called AdS Reissner-Nordström black holes. The extremal
Reissner-Nordström black holes have the property that their entropy is non-zero though
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the temperature vanishes 1 . This means that the extremal Reissner-Nordström black hole
violates the Nernst law of thermodynamics which states that for zero temperature the
entropy is typically also zero. From the AdS/CFT perspective then the dual field theory
displays the same behavior, i.e. also on the field theory side the entropy is non-zero at
vanishing temperature. This means that many different states are still accessible at T = 0
(cf. e.g. [8]), and the interpretation of that is not clear.
The natural question that arises is if there exist black solutions that fulfill the Nernst
law. In chapters 5 and 6 we construct such solutions within N = 2 U (1) gauged supergravity in four and five dimensions. It is important to mention that one of the features of
the solution space of these theories is the existence of horizons with non-spherical topology, such as R2 . These solutions are called black branes. We obtain extremal black brane
solutions with zero entropy density in four and five dimensions. We call these solutions
”Nernst branes” because they fulfill the aforementioned Nernst law.
In chapter 5 we consider N = 2 U (1) gauged supergravity in four dimensions with only
vector multiplets.
At first we study the equations of motion. The presence of the attractor mechanism
halves the degrees of freedom and hence the flow equations are given as ordinary firstorder differential equations. In practice, this means that we rewrite the action in terms of
squares of first order flow equations. Here we find it convenient to introduce a combination
of charges and fluxes whose phase is encoded in a parameter γ (see chapter 5 for the
technical details). We also obtain a flow equation for this phase parameter γ. Then we
proceed with exploring the solution space of the flow equations.
We first choose the simplest prepotential2 encoding one complex vector multiplet scalar
field, given by F = − i X 0 X 1 , and obtain a class of solutions representing non-Nernst
black branes. These turn out to be generalizations of the solution discussed in [9]. This
serves as a consolidating check on the formalism. Furthermore, we show a full numerical
solution that interpolates between a near-horizon AdS2 × R2 geometry and an asymptotic
1 3
AdS4 geometry for the prepotential F = − (XX 0) . All these solutions have constant γ. We
also find solutions with flowing γ but these turn out to be singular. It might be possible,
though, that they describe the asymptotic region of a global solution once higher derivative
corrections are taken into account.
We finally pursue the question of finding Nernst brane solutions. For this purpose, we
1 2X3
consider the STU model which is based on the prepotential F (X) = − X X
. We only
X0
look at solutions for which the physical scalars have vanishing real part. We find a solution
with constant γ and a fixed point at the zero of the radial coordinate. At this point, one
of the scalars flows to zero. Moreover, at this point, the metric has a coordinate singularity and the area density in the constant time and constant radial coordinate hyperplane
vanishes. The geometry near this fixed point has an infinitely long radial throat which
1
This is also true for the charged black holes in flat space and not only in asymptotically AdS space.
You find more information on Reissner-Nordström black holes in the introductory chapters 2 and 3.
2
In chapter 2 it is shown how prepotentials arise within N = 2 supergravity.
6
suppresses fluctuations in the scalar fields such that their solutions become independent
of their asymptotic values. We will take this infinite throat property to mean that the
solution is extremal (cf. chapter 2), and the vanishing of the area density to mean that the
solution has zero entropy density. However, these solutions asymptote to geometries that
are not AdS4 and, thus, it is not clear which role they might play in the gauge/gravity
correspondence.
Because of these difficulties in finding Nernst geometries with AdS asymptotics in four
dimensions, we then shift focus to gauged supergravity in five dimensions with the hope of
finding Nernst solutions in asymptotically AdS5 space.
Chapter 6 is devoted to the construction of five-dimensional black solutions with flat horizon in N = 2 U (1) gauged supergravity. As in four dimensions we begin by rewriting the
action in terms of squares of first-order flow equations. We also check that it is possible to
derive the first order equations from a superpotential. In addition the relation between the
five-dimensional first order equations and the four-dimensional ones obtained in chapter 5
is discussed.
The solutions we construct include Nernst solutions in asymptotic AdS5 backgrounds
as well as non-Nernst black brane solutions that describe extremal BTZ×R2 -solutions.
We find exact solutions with constant scalar fields that may have magnetic fields and
rotation. These solutions do not carry electric charge. We construct extremal BTZ×R2
solutions that are supported by magnetic fields, as well as rotating Nernst geometries in
asymptotic AdS5 backgrounds. Then we obtain numerical solutions with non-vanishing
scalar fields, with and without rotation. These have BTZ×R2 near horizon geometry and
are asymptotically AdS5 . They constitute generalizations of a solution given in [10] to
the case with several running scalar fields and rotation. We can immediately compute the
entropy density of the BTZ×R2 black brane by using the Cardy formula of the dual CFT.
A salient aspect of the first-order rewriting that gives rise to these black branes is the fact
that the angular momentum, the electric quantum numbers and the magnetic fields are
organized into quantities which are invariant under the spectral flow of the theory (cf. [11]
for the ungauged case).
In appendix E we turn to a different first-order rewriting in five dimensions. This is
motivated by the search for solutions with electric fields. This rewriting is based on the one
performed in [12] for static black hole solutions, which we adapt to the case of stationary
black branes in the presence of magnetic fields. The resulting first-order flow equations
allow for the non-extremal black brane solutions constructed in [13], as well as for the
extremal electric solutions obtained in [14, 10]. So far we could not obtain any dyonic
solution.
Chapter 2
Black Holes and Attractors
“Now there’s a look in your eyes like black holes in the sky...”
Pink Floyd, “Shine on you crazy diamond”
As the quotation shows the terminus “black hole” has meanwhile become part of the
popular culture. Indeed it is not surprising that these physical objects generate interest
also outside the scientific community. Losely speaking, a black hole is an object whose
gravitational forces are so strong that not even light can escape from it. Everything that
passes a special surface, the event horizon, is doomed to fall into a spacetime singularity.
On the way to the singularity the gravitational tidal forces will become larger and larger so
that finally everything is torn into pieces – no wonder that this dramatic and frightening
picture has made its way out to the public and did not stay hidden behind the “horizon”
of science.
However, a lot of work has been done in this fruitful field of research from experimental
physics even to pure mathematics. It is known that most galaxies have a black hole in their
centre whose mass is typically about 0.1 per cent of the mass of the galaxy. Recently it was
discovered that in the centre of the galaxy NGC 1277 there is a black hole that contains 59
per cent of the galaxy’s mass which is very unusual [15]. Apart from the experimentalists’
search for observational data concerning real existing black holes, black holes provide an
interesting field for theorists to work on. We will now give some background for the
theoretical and mathematical description of black holes.
The first black hole solution to the vacuum Einstein equations Rµν − 12 gµν = 0 was
found by the German physicist Karl Schwarzschild [16]. We will use this solution as a toy
model to present important concepts of black holes. All this can be found in textbooks on
general relativity like e.g. [3, 17, 18]. The line element for a Schwarzschild black hole with
mass M is given by
�
�
�
�−1
2M
2M
2
2
ds = − 1 −
dt + 1 −
dr2 + r2 dΩ2
(2.1)
r
r
with time coordinate t, radial coordinate r and dΩ2 the area element of the unit two-sphere.
At r = 2M the metric becomes singular, since gtt vanishes and grr diverges. This surface
8
2. Black Holes and Attractors
where gtt vanishes is called the event horizon. So what happens to an observer moving
towards the event horizon? The answer is: Nothing special!
It turns out that the singularity is merely a coordinate singularity which can be removed by choosing appropriate coordinates. This means that an observer passing the
event horizon will not die from very large tidal forces – at least not yet. However, this
does not mean that the event horizon is not useful for our description of black holes. The
horizon is indeed a surface which is special, and it can also be defined independently of
the chosen coordinate system. For r > 2M surfaces of constant r are timelike while for
r < 2M they are spacelike. For r = 2M it is a null surface. This is an important feature
to characterize the horizon independently of the chosen coordinates: A horizon is a null
surface. Moreover, the horizon does not change in time – it is static. More generally, for a
surface to be an event horizon it is even enough that it is stationary, i.e. invariant under
∂
the timelike Killing vector ∂t
. The latter case includes also Kerr black holes which are
rotating. To summarize, a horizon is a stationary null surface (cf. e.g. [19, 20] which are
nice reviews of many of the concepts to be discussed here). We use the conventions of [19].
The line element (2.1) has another singularity at r = 0. In contrast to the coordinate
singularity at r = 2M , the one at r = 0 cannot be removed by changing coordinates.
It is really a singularity of the spacetime. For r tending to zero the curvature of the
spacetime keeps on growing and the tidal forces diverge. Fortunately, this really bad
singularity is hidden behind the event horizon. What happens inside the black hole has no
effect on the world outside. This can be seen when one considers a light ray propagating
radially outwards from the black hole horizon. An observer far away from the black hole
would state that it takes an infinite amount of time for this light ray to reach him1 . But
this means that no information from the inside of the black hole can reach an observer
outside the black hole. Such an observer does not need to worry about the spacetime
singularity. This statement is governed by the cosmic censorship conjecture: There are no
naked singularities, i.e. spacetime singularities without an event horizon hiding them in
our universe (see e.g. [22] for a discussion).
After having defined the horizon of a black hole, let us introduce one further quantity,
the surface gravity. In Newtonian mechanics this is just the gravitational force a unit
test mass feels at the surface of an object. But since Newtonian mechanics is not the
appropriate theory to describe black holes, a different definition of the surface gravity is
needed. Consider an observer at infinity and a unit test mass closer to the black hole. The
observer at infinity will measure a certain force that is needed to keep the test mass at
rest. The surface gravity is then the limit of this force when the test mass approaches the
horizon.
For the Schwarzschild black hole the surface gravity is simply given by (cf. e.g. [20])
1
.
(2.2)
4M
A mathematically rigorous definition of the surface gravity for the case of an arbitrary
stationary black hole can be found e.g. in [17].
κ=
1
Calculations can be found e.g. in [21].
9
Schwarzschild’s black hole solution is of course not the only one. It is the simplest
possible black hole solution since it depends only on one parameter, namely the mass
of the black hole. But there are other black hole solutions that are charged and / or
rotating. Charged black holes are described by the Reissner-Nordström metric. We will
focus on them a bit later. Rotating black holes are the Kerr black holes we have already
mentioned. They give rise to interesting research topics even in mathematics (see e.g. [23]
for a mathematical introduction to this topic). For instance, S.T. Yau et. al. examined
the stability of the Kerr black hole against perturbations of spin s ∈ {0, 12 , 1, 2}. There are
rigorous mathematical proofs concerning scalar perturbations which are based on methods
from functional analysis and differential equations. Yau et. al. were able to prove that
solutions to the scalar wave equation in the Kerr geometry decay rapidly enough which
shows stability of the Kerr black hole. An overview of their results also concerning higher
spin perturbations can be found in [24] and the references therein.
As was already mentioned in the introduction, another remarkable feature of black holes
is that the laws of black hole mechanics are very similar to the laws of thermodynamics.
These ideas were presented for the first time in [25].
• Zeroth law: A body in thermodynamical equilibrium has constant temperature. The
analog for black holes is that the surface gravity is constant at the horizon of a
stationary black hole.
• First law: This is energy conservation. In thermodynamics the first law reads
dE = T dS + µdQ + ΩdJ .
(2.3)
For black holes we have
dM =
κ
dA + µdQ + ΩdJ .
8π
(2.4)
Here Q is the charge, µ the chemical potential, Ω the angular velocity and J the
angular momentum. One can see that the surface gravity κ plays the role of temperature which is not surprising in view of the zeroth law. The area of the black hole
horizon is in correspondence with the entropy which is also stated in the second law.
• Second law: In a thermodynamic process the total entropy S can never decrease,
∆S ≥ 0. Similarly, the area A of a black hole can never decrease, ∆A ≥ 0. This
means it is impossible that a black hole splits into two smaller ones. By contrast the
reversed process is not forbidden by the laws of black hole mechanics.
Are these merely analogies or is there a deeper connection between thermodynamics and
black holes? To answer this question let us consider a system with large entropy like the
desk of a hard working physicist with a lot of papers lying on it. What happens when
we throw this desk into a black hole? Then the entropy of our universe would decrease
in contradiction to classical thermodynamics. From considerations like this Bekenstein
concluded that black holes have to have entropy to ensure that the entropy in the universe
10
cannot decrease [26]. Having energy and entropy a black hole also has a temperature (cf.
e.g. [20]) by virtue of
1
∂S
=
.
(2.5)
T
∂E
Stephen Hawking showed in [27] that the temperature of a black hole is related to its
surface gravity precisely by
κ
T =
.
(2.6)
2π
From all this the famous formula for the Bekenstein-Hawking entropy of a black hole
follows:
A
S= .
(2.7)
4
Now that we have learned about horizons and the laws of black hole mechanics we want
to discuss the four dimensional Reissner-Nordström black hole in asymptotically flat space
in more detail. This will provide us with tools we need for the later consideration of more
complicated cases. We will follow [21] and [20]. The Reissner-Nordström black hole is a
solution to the equations of motion derived from the Langrangian density
�
�
√
1
µν
L = −g R − Fµν F
(2.8)
4
with the electromagnetic field strength given by
F = P sin θdθ ∧ dφ +
Q
dt ∧ dr .
r2
(2.9)
Here Q and P are the electric and magnetic charge, respectively. The line element for the
Reissner-Nordström black hole reads
�
�
�
�−1
2M
2M
P 2 + Q2
P 2 + Q2
2
ds2 = − 1 −
+
dt
+
1
−
+
dr2 + r2 dΩ2 .
(2.10)
r
r2
r
r2
One can see immediately that in the uncharged case, P = Q = 0, this reduces to the
Schwarzschild solution. Just like the Schwarzschild metric also the Reissner-Nordström
metric contains a spacetime singularity at r = 0. In contrast to the Schwarzschild case
now we have in general two coordinate singularities since gtt contains a quadratic term in
r. This yields two horizons, an inner and an outer one with radii given by
�
r± = M ± M 2 − (P 2 + Q2 ) .
(2.11)
The event horizon is then located at the larger root, i.e. at r = r+ . We only get real
solutions for M 2 ≥ P 2 + Q2 , otherwise there would be solutions with an imaginary part
different from zero. The latter correspond to naked singularities which are considered
unphysical due to the cosmic censorship conjecture. So we obtain that the mass of the
Reissner-Nordström black hole is bounded from below. For M 2 = P 2 + Q2 there is only
one real solution and the two horizons coincide. Such black holes are called extremal .
11
Extremal black holes have a number of interesting properties. The temperature and
entropy of the Reissner-Nordström black hole are given by (cf. e.g. [20])
�
M 2 − (P 2 + Q2 )
�
T =
,
2π(2M (M + M 2 − (P 2 + Q2 )) − (P 2 + Q2 ))
�
2
S = πr+
= π(M + M 2 − (P 2 + Q2 ))2 .
(2.12)
One can see immediately that the temperature of an extremal Reissner-Nordström black
hole is zero. This is also true in more general cases than Reissner-Nordström. Extremal
black holes have vanishing temperature and because of this they are thermodynamically
stable at the classical level. Further, the entropy of the Reissner-Nordström black hole is
non-zero at the horizon and its value is determined only in terms of the black hole charges.
Furthermore the horizon of an extremal black hole lies at the end of an infinitely long
throat. To see this we need to have a closer look at the near horizon geometry of the
Reissner-Nordström black hole. With the coordinate change
ρ=r−M
the extremal Reissner-Nordström black hole takes the form
�
�−2
�
�2
M
M
2
2
ds = − 1 +
dt + 1 +
(dρ2 + ρ2 dΩ2 ) .
ρ
ρ
(2.13)
(2.14)
These coordinates are called isotropic (cf. e.g. [20, 28]). Now the tt-component of the
metric has a double zero at ρ = 0 and therefore in isotropic coordinates the horizon is
located there. Near the horizon, i.e. for ρ → 0, the line element can be written as
ds2 = −
ρ2 2 M 2 2
dt + 2 dρ + M 2 dΩ2 .
M2
ρ
(2.15)
One can see that in this particular limit the spacetime is a direct product of a two-sphere
of radius M and a two-dimensional spacetime with metric element
ds̃2 = −
ρ2 2 M 2 2
dt + 2 dρ .
M2
ρ
(2.16)
This is the line element of the two-dimensional Anti-de-Sitter space AdS2 . Anti-de-Sitter
spaces play an important role in string theory and we will give further information on them
in section 3.2. But let us have a closer look at the line element (2.16). The radial distance
d of an observer at ρ0 > 0 to the horizon is (up to overall constant prefactors) determined
by (cf. e.g. [28])
� ρ0
1
d ∼ lim
dρ = lim (log(ρ0 ) − log(�)) = ∞.
(2.17)
�→0 �
�→0
ρ
In this calculation we assumed that is it possible to use the near-horizon approximation
(2.16). We see, that the horizon is infinitely far away from the observer.
12
✻
radial direction
Figure 2.1: Non-extremal and extremal black hole. The extremal one on the right displays
an infinitely long throat.
But this means that the horizon lies at the end of an infinite throat (cf. e.g. [20]). An
analogous calculation shows that the situation is different for the non-extremal ReissnerNordström black hole. Here the event horizon is at finite distance from an observer at
finite ρ0 (cf. [21]). You find an illustration of the two different situations in figure 2.1.
This property of having a throat geometry will be very important in chapter 5 where we
will use this property to distinguish whether a black solution is extremal or not.
In chapter 5 and 6 we will also look at horizon geometries including an AdS2 factor
but with horizon topology AdS2 × R2 instead of AdS2 × S 2 . Such black solutions with
non-spherical but flat horizon topology are called black branes. Such objects cannot exist
in asymptotically flat space as is known from statements on the uniqueness black hole
constructions (cf. e.g. [29, 30, 31]). In a space with asymptotic AdS behavior it is well
possible for a black object to have non-spherical horizon topology. Therefore black branes
are especially important in the context of the AdS/CFT correspondence.
2.1
The attractor mechanism
For our purpose of studying black brane solutions another interesting feature of extremal
black configurations is important, namely the attractor mechanism. Early work on this
subject can be found for instance in [32, 33, 34, 35, 36, 37, 32, 38]. An overview is given
e.g. in [21]. In general, to study the attractor mechanism black solutions in supergravity
are considered. In chapters 5 and 6 we will look at the four- and five-dimensional case
of N = 2 supergravity, so we will restrict ourselves to these particular dimensions. For
2.1 The attractor mechanism
13
the introduction of the attractor mechanism we will consider the four-dimensional case
for simplicity. The chosen supergravity theory always contains a set of scalar fields, and
the general idea of the attractor mechanism is that at the horizon of the black hole the
scalar fields included in the theory flow towards a value that is independent of the initial
conditions far away from the horizon. This value is completely determined by the charges
of the black hole. In the following we will present the basic features of the attractor
mechanism. Here we stick closely to [21, 39]. The scalar fields of the supergravity under
consideration typically have an effect on the couplings of the vector fields in the theory.
The bosonic part of an action describing such a theory in four dimensions is given by (cf.
e.g. [21])
�
�
√
1
1
1
�µνρσ I J
i µ j
I
J µν
L = −g R − gij (φ)∂µ φ ∂ φ + IIJ (φ)Fµν F
+ RIJ (φ) √ Fµν Fρσ . (2.18)
2
4
4
2 −g
Here I, J = 0, . . . , n denotes the number of vector fields and i, j = 1, . . . , nS the number
of scalars. These numbers are specified when choosing a specific model. The matrix I is
negative definite and describes the gauge kinetic couplings and it depends on the scalar
fields φi . The matrix R is a generalization of the θ-angle terms (cf. e.g. [21]). In a more
general case one can also look at gauged supergravity. Then there is an additional scalar
potential in the Lagrangian as we will see in chapters 5 and 6. For the purpose of explaining
the attractor mechanism we can set the scalar potential to zero.
The scalar fields can be viewed as coordinates on a manifold with metric gij . In four
dimensions and N = 2 supergravity this manifold has some interesting features: It is a
special Kähler manifold. We will address this point later and give a precise definition and
some properties of special Kähler spaces.
But for the moment the metric can be kept arbitrary and we will turn to the more
specific case of N = 2 supergravity later.
In [21] an ansatz is made for the metric of a static spherically symmetric and charged
black hole solution, namely
ds2 = −e2U (r) dt2 + e−2U (r) dr2 + r2 dΩ2
(2.19)
with dΩ2 the line element of a two-sphere. Since all quantities U and φi are assumed
to depend only on the radial variable r the action (2.18) can be reduced to an effective
1-dimensional action [21]
1
L1d = (U � )2 + gij (φi )� (φj )� + e2U VBH .
2
(2.20)
Here VBH is the so-called black hole potential which is given by [40]
1
VBH = − QT MQ
2
with the matrix
M=
�
I + RI −1 R −RI −1
−I −1 R
I −1
(2.21)
�
(2.22)
14
and Q the charge vector
� I�
p
Q=
qI
obtained from the vector field strengths and their duals by
�
�
1
1
I
I
p =
F , qI =
GI .
4π S 2
4π S 2
(2.23)
(2.24)
The sphere S 2 is taken to be at infinity and the dual field strength GI is related to F I
through [21]
δL
GI = − I = RIJ F J − IIJ � F J .
(2.25)
δF
The equations of motion for the metric function U and the scalar fields are
U �� = e2U VBH ,
(φi )�� + Γjki (φj )� (φk )� = e2U g ij ∂j VBH .
(2.26)
Here Γijk are the Christoffel symbols. To make sure that solutions to the equations of
motion coming from the 1-dimensional effective action solve also the original 4-dimensional
equations of motion one has to implement the Hamiltonian constraint (cf. e.g. [21])
1
H = 0 ⇐⇒ (U � )2 + gij (φi )� (φj )� − e2U VBH = 0 .
2
(2.27)
At this point we are able to establish the attractor mechanism. The equations (2.26)
describe a dynamical system in the radial variable r, and thus one can ask if they give
rise to an attractor. To define an attractor in a rigorous mathematical way, one has to
convert the given dynamical system to a system of first-order differential equations, which
is always possible. This means one writes the higher order differential equations in the form
d
�x(r) = f�(�x(r), r) ( cf. e.g. [41]). Here �x is the vector (U, U � , φ1 , (φ1 )� , . . . )T . The dots
dr
stand for the other functions that occur in the given system of second-order differential
equations. f� describes how the system of first-order differential equations explicitly looks
like. It may also depend explicitly on r.
With the notion of the flux Φr which maps the initial conditions �x0 to �x(r) one can
define an attractor in the following way.
Definition. An attractor is a compact subset A of the domain of definition of the function
f� such that
1. A is invariant under the flux Φr , i.e. Φr (A) ⊂ A and
2. there exists an open neighbourhood U of A that is invariant under Φr and has the
additional property that for any neighbourhood V of A with V ⊂ U there is a value
rV such that Φr (U ) ⊂ V for any r > rV .
15
�10 �5
0
5
10
40
30
20
10
20
10
0
0
�10
Figure 2.2: The Lorenz attractor
Furthermore, it is often postulated that the attractor set A is connected [41]. The second
property explains the “attractivity” of the set A. Points close enough to A are mapped
by Φr to points that are even closer to A. This means that while the system evolves the
points in U finally reach the attractor A and can never escape from it again. The largest
such set U with the property mentioned in the second part of the definition is called the
basin of attraction. For initial conditions within this set the solution eventually flows to
the attractor set.
For general dynamical systems attractors can exhibit a very complicated structure, as
e.g. the Lorenz attractor does. It was found by the meteorologist Edward Lorenz when
he studied the convection and heat distribution on a rectangular area (cf. e.g.[42]). This
attractor is an example for a so-called strange attractor. It is a fractal, i.e. a geometrical
object with non-integer dimension and properties like self-similarity (cf. e.g. [42]). You
find a picture of the Lorenz attractor in figure 2.2.
On the other hand, attractors can also be very simple objects, as is the case for black
holes. Here the attractor is only a single point in the domain of definition for the radial
variable r. It is a fixed point of the dynamical system. The expression “fixed point” is to
be understood in the following way. We could consider the sequence of points {�x(rn )}n∈N
for appropriately chosen values2 rn . Then one can look at the map G that takes �x(rn )
to �x(rn+1 ). Reaching a point attractor means that the map G has a fixed point, i.e.
G(�x∗ ) = �x∗ and the attractor is only the single point �x∗ . Pointlike attractors occur e.g. in
the presence of a potential that displays a minimum. A very simple example is a damped
spring pendulum. The potential is quadratic in the amplitude and has a minimum at the
2
There are several techniques of how to choose this sequence, but since this is not that important for
understanding the main principles it will be omitted here.
16
equilibrium length l0 of the spring. For any initial condition the amplitude finally tends to
the equilibrium length l0 and the velocity of the pendulum mass goes to zero. In this case
the vector (x, x� )T = (l0 , 0)T is the attractor point.
Since in our case (2.26) there is also a potential we can ask if there is a similar mechanism
at work. This means actually that we have to look for minima of the black hole potential.
If we assume that the black hole potential has a minimum at φi∗ then we can conclude from
the second equation in (2.26) that φi (r) = φi∗ = const is a solution �to the equations
� � i∗of
�
i
φ
φ
(r)
i
motion for the φ since all derivatives vanish in this case. this means
=
(φi )� (r)
0
i∗
is a fixed point of the dynamical system. Since we chose the φ such that the potential is
minimized we can conclude that the fixed point is an attractor just like in the case of the
spring pendulum mentioned before.
Apart from the dependence on the scalar fields φi the only expressions that enter the
black hole potential are the electric and magnetic charges. This implies that the values φi∗
depend only on the charges of the black hole. This is also a way to characterize the horizon
of an extremal black hole: The horizon lies at that radial point where the critical values
φi∗ are reached that minimize the black hole potential (cf. e.g. [21]). To summarize, the
scalar fields are driven to constant values at the horizon that are determined completely
by the charges of the black hole and that do not depend on the initial conditions for the
φi (here it is assumed that the basin of attraction corresponds to the maximal domain of
definition for the φi , the appearance of different basins of attraction was e.g. discussed in
[43, 44]).
The question that naturally arises at this stage is how do we know about the existence
of minima of the black hole potential? An answer is given in [45]. To understand the main
argument in [45] we need to relate the black hole potential to the central charge Z of the
supergravity theory under consideration. We will not discuss the most general case but
instead only give the argument for N = 2 theories in four dimensions. In this case the
central charge of the underlying supersymmetry algebra is the related to the black hole
potential via (cf. e.g. [21])
VBH = |Z|2 + 4g i̄ ∂i |Z|∂¯̄ |Z|
(2.28)
with gi̄ the Kähler metric on the moduli space (see also section 2.3). In this case the more
general Lagrangian density (2.20) can be written in the form [21]
L = (U � )2 + gi̄ (z i )� (z̄ ̄ )� + e2U (|Z|2 + 4g i̄ ∂i |Z|∂¯̄ |Z|) .
(2.29)
From now on the scalars will be denoted by z i instead of φi to depict the restriction to
4-dimensional N = 2 supergravity. In [21] it is proven that a critical point of |Z|, i.e. a
point r∗ with ∂i |Z(r∗ )| = 0, is also a critical point of the black hole potential. Moreover, it
can be shown that values of the radial variable for which the central charge is minimized
also yield minima of the black hole potential. This means that the existence of minima
of the central charge implies the existence of minima of the black hole potential. In [45]
conditions for the existence of minima of the central charge are discussed. In theorem 2.5.1.
17
of [45] it is stated that a non-zero minimum of |Z|2 (which implies a minimum of |Z| since
the map |Z|2 �→ |Z| is strictly increasing) exists under certain conditions on the charges
of the black hole. This non-zero minimum of the central charge exists if the electric and
magnetic charges are integral and if there exists a number C ∈ C such that [45]
C̄X I − C X̄ I = ipI ,
C̄FI − C F̄I = iqI .
(2.30)
Here X I are coordinates on the so-called big moduli space and for the cases which are
important for us FI are dericatives of a function called the prepotential. At this point we
will not go into the details. We will provide more information on the X I and FI in section
2.3. For the moment the most important point in this discussion is that for a given set of
black hole charges with “good enough” properties we know that there is an attractor of
the flow equations located at the black hole horizon. Further, the attractor values of the
scalar fields and of the black hole potential are completely fixed by the black hole charges.
Another important feature of extremal black hole solutions is that they admit a description
in terms of first-order flow equations. In four-dimensional ungauged supergravity these
equations were first constructed and solved for black holes in [32, 33]. As mentioned
before, solutions to the equations of motion for the Lagrangian density (2.20) have to obey
the Hamiltonian constraint (2.27)
(U � )2 + gi̄ (z i )� (z̄ ̄ )� = e2U (|Z|2 + 4g i̄ ∂i |Z|∂¯̄ |Z|) .
(2.31)
A very simple ansatz to fulfill the Hamiltonian constraint is to equate the appearing squares
in an appropriate way:
U � = ±eU |Z| ,
(z i )� = ±2eU g i̄ ∂¯̄ |Z| .
(2.32)
It can be shown that also the second-order equations of motion are satisfied if one chooses
the same sign in both equations (cf. [21]). In order to reproduce the extremal ReissnerNordström black hole in flat space one has to choose the minus sign in (2.32). The extremal
Reissner-Nordström black hole has the interesting property of being supersymmetric. We
will comment on that later in this section. Thus, taking the minus sign in (2.32) leads to
supersymmetric solutions and results in the first-order flow equations
U � = −eU |Z| ,
(z i )� = −2eU g i̄ ∂¯̄ |Z| .
(2.33)
Moreover, the Lagrangian density (2.29) can be rewritten as a sum of these first-order flow
equations up to a total derivative:
L = (U � + eU |Z|)2 + gi̄ ((z i )� + 2eU g ik̄ ∂¯k̄ |Z|)((z̄ ̄ )� + 2eU g ̄l ∂l |Z|) + T D .
(2.34)
18
The variation of the Lagrangian density written in the form (2.34) vanishes if the first-order
differential equations (2.33) are satisfied. To summarize, the solutions to the first-order
equations fulfill the Hamiltonian constraint and therefore also the second-order equations.
Of course, the converse need not be true: There might be solutions to the second-order
equations that do not obey the first-order equations. Inspite of this there is a great advantage of having first-order flow equations instead of second-order ones. Generally speaking
first-order differential equations are easier to solve than second-order equations since we
only have to specify half the number of boundary conditions. In combination with the
attractor mechanism the appearance of first-order equations can be interpreted as follows.
As the horizon values for the scalars and their derivatives are completely determined by
the black hole charges the number of boundary conditions one needs to specify a solution
is already reduced. Therefore first-order equations provide enough information to fix a
solution completely.
As you can see in (2.33) the first-order equations in this example can be written in
terms of derivatives of the central charge. In more general cases it is possible to express
the first-order equations in terms of a so-called superpotential instead of the central charge.
Also in this case the flow equations are determined by derivatives of the superpotential.
We will see an example of a superpotential in chapter 6.
So far we have seen how the attractor mechanism works and also how to rewrite the
equations of motion as first-order differential equations.
Since the attractor mechanism ”halves” the number of free parameters (i.e. the number
of boundary conditions) we might ask if there is a connection to BPS states. BPS states
in supersymmetry are states that are invariant under a certain number (usually half or a
quarter) of supersymmetry transformations. We will have a closer look at BPS states in
the next section.
2.2
Black holes as BPS states
BPS states are very important in supersymmetry. Given a theory with N ≥ 1 supersymmetries there are N spinorial charges QA
α with A = 1, . . . N and α = 1, 2. Due to the
theorem of Haag, Lopuzanski and Sohnius (cf. [46]) the most general SUSY algebra is
then (cf. e.g. [47], [48])
µ
B †
AB
{QA
,
α , (Qβ ) } = 2σαβ Pµ δ
B
AB
{QA
.
α , Qβ } = �αβ Z
�
0 −1
and σ µ = (− 2 , σi ) with σi the Pauli
1 0
matrices. Z AB is the central charge matrix. We want to consider massive representations
in the case N = 2. Here we are left with only one central charge Z 12 . In the rest frame we
have Pµ = (−M, 0, 0, 0) and the algebra (2.35) simplifies to (with the notation conventions
Here Pµ is the generator of translations, � =
�
(2.35)
2.2 Black holes as BPS states
19
used in [47])
B †
AB
{QA
,
α , (Qβ ) } = 2M δαβ δ
B
12
AB
{QA
.
α , Qβ } = |Z |�αβ �
(2.36)
Via U (2) transformations applied to the QA
α the central charge matrix can be brought into
the form
�
�
0
2|Z|
12
Z =
(2.37)
−2|Z| 0
with 2|Z| = |Z 12 |. One can then define creation and annihilation operators by taking
appropriate linear combinations of the SUSY charges QA
α (cf. e.g. [48]):
�
1 �
aα = √ Q1α + �αβ (Q2β )† ,
2
�
1 � 1
bα = √ Qα − �αβ (Q2β )† .
2
(2.38)
These operators fulfill the anticommutation relations
{aα , a†β } = 2(M − |Z|)δαβ ,
{bα , b†β } = 2(M + |Z|)δαβ .
(2.39)
The left hand side of the first equation is a positive semi-definite operator, so also the right
hand side has to be positive semi-definite. This leads to the so-called BPS bound
M ≥ |Z| .
(2.40)
If M > |Z| we have 2N = 4 creation operators and one can create 22N = 16 states starting
from a state with minimal helicity. When the BPS bound is saturated, we have to set the
operators aα , α ∈ {1, 2} to zero and so one can only create only 22(N −1) = 4 different states
starting from a given helicity. This procedure can easily be generalized to theories with
more than two SUSY charges. Then the central charge matrix is block-diagonal with each
block of the form (2.37) and one defines creation and annihilation operators in analogy to
(2.38). For N > 2 there are more possibilities of how many creation operators have to
be set to zero in the BPS case. More precisely, it is possible that k ≤ N2 of the creation
operators have to be set to zero (cf. e.g. [48]). It is important to mention that BPS states
are the states with minimal mass for a given central charge, i.e. for a given set of quantum
numbers.
So far we have discussed BPS states in supersymmetry, but since we want to interpret black holes as BPS states, we need a notion of BPS states in a gravity theory and
the theory at hand is (not surprisingly) supergravity. Supergravity is the local version
of supersymmetry, i.e. all supersymmetry transformations that were so far described by
constant (spinorial) parameters � are now described in terms of parameters �(x) that depend on the spacetime coordinates. To understand this better we first have to take a
20
step backwards and look how SUSY transformations can be realized on fields. Here it is
useful to enlarge the space on which the SUSY fields live by anticommuting Grassmannian
coordinates θ and θ̄. This space with coordinates (xµ , θ, θ̄) (with xµ the usual spacetime
coordinates) is called superspace and fields depending on these coordinates are therefore
named superfields. The idea is now to realize the SUSY generators Qα and their conjugates
as differential operators on superspace. This means that i�α Qα is supposed to generate
a translation in the Grassmannian θ-direction by a constant infinitesimal spinor � and in
addition a translation in spacetime [48].The explicit form of the differential operator Qα is
(cf. [49])
∂
µ β
Qα = α − iσαβ
θ̄ ∂µ .
(2.41)
∂θ
Then the transformation law for a superfield F is [49]
δ� F (xµ , θ, θ̄) = (�Q + �¯Q† )F .
(2.42)
The operator �Q + �¯Q† generates a coordinate shift by the infinitesimal parameter �:
(xµ , θ, θ̄) �→ (xµ + iθσ µ �¯ − i�σ µ θ̄, θ + �, θ̄ + �¯) .
(2.43)
In supergravity the transformation parameters � may depend on the spacetime coordinates,
i.e. now one wants to take into account motions in superspace of the form [49]
(xµ , θ, θ̄) �→ (xµ + iθσ µ �¯(x) − i�(x)σ µ θ̄, θ + �(x), θ̄ + �¯(x)) .
(2.44)
One can see here that the infinitesimal shift in the coordinates depends on the spacetime
position xµ and so a coordinate transformation is generated that might be not only a translation. Since the theory dealing with coordinate transformations (i.e. diffeomorphisms) is
general relativity, one can see immediately that for having supersymmetry realized locally
one has to include gravity. Indeed, the gauge field for those coordinate transformations
is the graviton. Its superpartner, the gravitino, is the gauge field for the local SUSY
transformations (cf. e.g. [47]).
Also in supergravity BPS states are states which are invariant under a subset of SUSY
transformations, but these are now local transformations. Invariance means here the following. We can look at a classical solution to the equations of motion and consider the
variation of all fields under local supersymmetry. The solution is invariant under local
SUSY if the variation of all fields vanish when evaluating it in the classical background
solution. Schematically this can be expressed as
δ(�(x))boson = �(x)fermion = 0 ,
δ(�(x))fermion = �(x)boson = 0 .
(2.45)
We will not write down the explicit expressions for the field variations since this requires
lots of technical details. Instead we will restrict ourselves to a more qualitative description
and refer to [50] for details. On a classical solution all fermionic fields are zero anyway so
that the first condition is automatically satisfied. Setting the variation of all fermionic fields
2.2 Black holes as BPS states
21
to zero leads to conditions on the infinitesimal parameter �(x). For example, when taking
the simplest background solution, namely flat Minkowski space, �(x) has to be constant
[50]. In general, setting the variations of the fermions to zero, leads to a set of first-order
differential equations in the bosonic fields like the metric, scalars and vector fields and also
in the transformation parameter �(x). For example, the first-order equations (2.33) can be
obtained from setting the variation of the gravitino and the gaugino to zero (cf. e.g. [21]).
In the Minkowski example one gets for the variation of the graviton and the gravitino the
following equations [50]
1
δeaµ = �¯(x)γ a ψµ = 0 ,
2
δψµ = Dµ �(x) = 0 .
(2.46)
Here eaµ is the vielbein and ψµ denotes the gravitino. In classical four-dimensional Minkowski
space the gravitino field vanishes, so the first equation is automatically satisfied. The second one sets restrictions on the transformation parameter �(x). In Minkowski space the
solutions to the second equation in (2.46) are four independent constant Majorana spinors
�α , α = 1, . . . , 4 (cf. e.g. [50]). In the general case a solution for the transformation parameters is built of some number k of independent spinors �α (x) with α = 1, . . . , k. These
independent spinors are called Killing spinors and each of them stands for a preserved
supersymmetry. They are the analogon of Killing vectors in general relativity. Just as the
latter describe isometries of the spacetime, Killing spinors decribe the preserved supersymmetries. When k of the original N supersymmetries are preserved, the solution is called
k
-BPS.
N
It was discussed in [45] under what conditions on the central charge BPS states exist.
There are three distinct cases:
1. |Z|2 has a non-zero minimum. Then BPS states are expected to exist. As already
mentioned the conditions on the black hole charges for the existence of a nonvanishing
minimum are also examined in [45].
2. |Z|2 does not have a stationary point on the interior of its domain of definition. It
might vanish at the boundary. Then the supergravity approximation is not valid and
it is not possible to decide whether BPS states exist or not.
3. For some choice of charges the minimal value of |Z|2 can be zero. Then there are no
BPS states in the theory.
An example of a BPS state�in supergravity is the extremal Reissner-Nordström black
hole in flat space. Here |Z| = P 2 + Q2 (cf. e.g. [51]) and the extremal black hole saturates the BPS bound (cf. the discussion below (2.11)). This is only true in asymptotically
flat space. In section 3.5 we will consider the Reissner-Nordström black hole in asymptotically AdS space. In this case the mass of the extremal black hole is strictly larger than the
BPS mass bound, i.e. the extremal Reissner-Nordström black hole with AdS asymptotics
is not supersymmetric. The BPS solution on the other hand displays a naked singularity
(cf. [8]).
22
Now that we have seen how the attractor mechanism works and how supersymmetry enters
the discussion of black holes we want to come back to what we mentioned at the beginning
of this section, namely the structure of the scalar manifold.
2.3
Special geometry
Before we will study the geometry of the scalar manifold in N = 2, D = 4 supergravity
we review some features of the vector fields in the action (2.18) and their duals (2.25).
First of all, generically there are three main types of multiplets in this theory. There is
the gravity multiplet containing the graviton and two gravitini as well as a vector field
called graviphoton. Then there are n vector multiplets, each equipped with a vector field,
a gaugino and a complex scalar. In addition, there is a number of hypermultiplets which
also contain scalars. For simplicity we do not consider hypermultiplets here. Since we have
n vector multiplets each containing a vector field and in addition the graviphoton there
are n + 1 vector fields.
The equations of motion and the Bianchi identities for the vector fields are (cf. [21])
dF I = 0 ,
dGI = 0 .
(2.47)
From this one can see immediately that a transformation
� �
� ��
� �
F
F
F
�→
=S
G
G�
G
(2.48)
with a matrix S ∈ GL(2(n + 1), R) leaves the equations of motion and Bianchi identities
invariant because of the linearity of the derivative. When the transformation is performed
also the Lagrangian changes and one has to take care that expression for the transformed
dual field strength G�I is consistent with its definition according to (2.25). This requirement
restricts the matrix S to be an element of the symplectic group Sp(2(n+1), R) ⊂ GL(2(n+
1), R), i.e. S has to obey the equation
�
�
�
�
0
0
n+1
n+1
T
S
S=
.
(2.49)
− n+1
0
− n+1
0
Writing S =
A, B, C, D:
�
A B
C D
�
in blocks of size (n + 1) × (n + 1) yields the following conditions on
AT C − C T A = 0 = B T D − DT B ,
AT D − C T B =
n+1
.
(2.50)
Details of this calculations can be found e.g. in [52]. There are some comments in order
at this point. Firstly, due to additional matter couplings in the Lagrangian there might
be more restrictions on the transformation matrix such that it has to be an element only
of a subgroup of the group of symplectic matrices. This subgroup (which of course can
2.3 Special geometry
23
be identical with the full group of symplectic matrices) is called the U-duality group. Secondly, the stress-energy tensor is not changed under U-duality rotations and so neither
the Einstein equations nor the metric are modified. This means that one can start with a
solution to the equations for the Maxwell fields, i.e. a black hole solution with a fixed set of
electric and magnetic charges, and perform a U-duality rotation to generate new solutions
with different charges from the known one without changing gravitational properties (cf.
[21]). Thirdly, it is important to mention that U-duality is in general not a symmetry
of the Lagrangian, i.e. the Lagrangian will usually change its form when performing a
U-duality transformation. This statement is encoded in the consistency requirement on G�I
when deriving it from the Lagrangian which was already mentioned above.
But what happens now to the scalar fields when one performs a duality rotation among the
vector fields? The coupling matrices I and R in the Lagrangian (2.18) are also changed
under a duality transformation and since they depend on the scalar fields the scalars get
modified, too. This can be understood as follows. The scalars can be interpreted as coordinates on some manifold Mscalar and a duality rotation of the vector fields leads to a
transformation of the scalars by a diffeomorphism of the scalar manifold. This means that
the diffeomorphisms on the scalar manifold are somehow related to the duality transformations of the vector fields [52]. Of course, this sets restrictions on the manifolds that come
into consideration for Mscalar .
In the case N = 2, D = 4 the scalar manifold can be written as a Cartesian product of
manifold for the scalars sitting in the hypermultiplets and a different one for those sitting
in the vector multiplets. So we have
Mscalar = Mv × Mh
(2.51)
with Mv/h the manifold for the scalars in the vector multiplets or the hypermultiplets
respectively. Because of this factorized structure of the scalar manifold and since the
scalars in the hypermultiplets are not important for our discussion, we will only explain
the properties of Mv in more detail.
As we already mentioned at some point, the manifold for the complex scalars in the
vector multiplets in our case is a Kähler manifold. But not only this, it also has some
additional properties which make the the manifold even more special - indeed its geometry
is called “special geometry”.
But let us first define a Kähler manifold and then look at what provides it with special
geometry. The definitions for Kähler manifolds are taken from [50] and [53]. But before
we come to that, we have to make some preparing definitions.
A complex manifold has a generic line element of the form
ds2 = 2gαβ̄ dz α z̄ β̄ + gαβ dz α dz β + gᾱβ̄ dz̄ ᾱ dz̄ β̄ .
(2.52)
A metric on a complex manifold is called Hermitian if there exist coordinates in which
gαβ = gᾱβ̄ = 0
(2.53)
24
so that the line element takes the form
ds2 = 2gαβ̄ dz α z̄ β̄ .
(2.54)
The fundamental 2-form for a Hermitian metric is defined as
K = −2igαβ̄ dz α ∧ dz̄ β̄ .
(2.55)
Now we can define a Kähler manifold.
Definition. A Kähler manifold is a complex manifold with Hermitian metric whose fundamental 2-form is closed, i.e. dK = 0. In this case the fundamental 2-form is called the
Kähler form.
That the Kähler form is closed is equivalent to the condition that
∂γ gαβ̄ − ∂α gγ β̄ = 0
(2.56)
which means that in every coordinate patch the metric can be expressed through the second
derivatives of a function K called the Kähler potential:
gαβ̄ = ∂α ∂β̄ K(z, z̄) .
(2.57)
The Kähler potential is not determined uniquely since the potential
K � (z, z̄) = K(z, z̄) + f (z) + f¯(z̄)
(2.58)
yields the same metric as the original potential K. In fact, in the overlapping regime of two
coordinate patches, the different Kähler potentials are related by a transformation as in
(2.58) which are thence called Kähler transformations. Often Kähler manifolds are defined
in a different way which we will present shortly. For this purpose we need two additional
definitions.
Definition. A homomorphism of vector bundles J : T M → T M is called an almost
complex structure on the manifold M, if J 2 = −IdT M .
The prefix ”almost” indicates that the manifold M need not be equipped with a complex
structure although the tangent spaces Tx M exhibit a complex structure.
Definition. A Riemannian manifold M with metric g and orthogonal almost complex
structure J, i.e. with
g(JX, JY ) = g(X, Y )
(2.59)
for all vector fields X, Y , is called an almost Hermitian manifold.
Now the following theorem can be stated.
25
Theorem. A manifold is a Kähler manifold if and only if it is an almost Hermitian
manifold whose almost complex structure J is parallel, i.e. for which ∇g J = 0. Here ∇g is
the Levi-Civita-connection.
Now that we know what a Kähler manifold is we can proceed and turn to special geometry
(see e.g. [54, 55, 56] for mathematical approaches to special geometry). There are two
types of Kähler manifolds which are called special manifolds. On the one hand, there are
the so-called “rigid” or “affine” special Kähler manifolds. They arise in the context of rigid
N = 2 supersymmetry. Their local counterpart is called “projective“ special geometry or
often only special geometry. In the following we will give definitions for special geometry
either in the rigid and in the local case. For this purpose we will rely on [57]. There the
authors display different definitions for special geometry and show their equivalence.
Generally speaking, in special geometry the scalars are affected by the symplectic transformations for the vector field strengths. To see how this works let us have a look at rigid
special geometry first.
Definition. An n-dimensional Kähler manifold is a rigid special Kähler manifold if the
following conditions are satisfied:
1. On every coordinate chart there are n independent holomorphic functions X I (z), I =
1, . . . , n and a holomorphic function F (X) called the prepotential such that the
Kähler potential takes the form
�
�
I ∂
I ∂
K(z, z̄) = i X
F̄ (X̄) − X̄
F (X) .
(2.60)
∂X I
∂ X̄ I
2. On the intersection of two charts Ui ∩ Uj �= ∅ there are transition functions of the
following form:
� �
� �
X
X
icij
= e Mij
+ bij
(2.61)
∂F (i)
∂F (j)
with cij ∈ R, Mij ∈ Sp(2n, R) and bij ∈ C2n .
3. The transition functions satisfy the cocycle condition, i.e. on the intersection of three
charts Ui ∩ Uj ∩ Uk �= ∅
ecij ecjk ecki = 1 ,
Mij Mjk Mki =
2n
.
(2.62)
Equivalently, a rigid special Kähler geometry can be defined as follows.
Definition. A manifold M is a rigid special Kähler manifold if
1. there exists a U (1) × ISp(2n, R)3 vector bundle over M with constant transition
functions as in (2.61) and a holomorphic section V such that the Kähler form can be
3
The group ISp(2n, R) is defined as ISp(2n, R) = Sp(2n, R)�T (2n, R), i.e. it is the semi-direct product
of the symplectic group Sp(2n, R) and the translations on R2n . ISp(2n, R) is called the inhomogeneous
symplectic group (cf. [58]).
26
written as
1 ¯
∂ ∂�V, V �
2π
with the symplectic inner product defined as
�
�
0
n
T
�V, V � := V
V.
− n 0
K=−
(2.63)
(2.64)
and
2. if in addition
where ∂α =
∂
.
∂z α
�∂α V, ∂β V � = 0
(2.65)
From the second definition we see the connection to the symplectic transformations we
discussed before. In [57] also a third equivalent definition is given, but we restrict ourselves to the two definitions displayed above since the main ingredients we need for further
applications can already be seen from them. These main features are the prepotential and
the underlying symplectic structure.
Now that we explained the structure of rigid special geometry we come to the more
general case of local special geometry. From here on we will use the term ”special geometry“
to describe local special geometry. In the local case the Kähler manifold is required to be a
Kähler-Hodge manifold. This means that its Kähler form has to be of integer cohomology.
Such manifolds are also called ”Kähler manifolds of restricted type“ or simply ”Hodge
manifolds“. This additional condition on the Kähler manifold comes from the fermions
which impose a quantization condition on the Kähler form (cf. [57]).
At this point we are ready to define special Kähler manifolds. We will again show two
equivalent definitions.
Definition. An n-dimensional Kähler-Hodge manifold is called special Kähler if it has the
following properties:
1. On every chart there exist projective coordinate functions X I (z), I = 0, . . . , n and
a holomorphic function F (X) which is homogeneous of degree two, such that the
Kähler potential can be written as
�
�
I ∂
I ∂
K(z, z̄) = − log iX̄
F (X) − iX
F̄ (X̄) .
(2.66)
∂X I
∂ X̄ I
F is the prepotential.
2. On the intersection of two charts Ui ∩ Uj �= ∅ there are transition functions of the
following form:
� �
� �
X
X
fij (z)
=e
Mij
(2.67)
∂F (i)
∂F (j)
with holomorphic functions fij and Mij ∈ Sp(2(n + 1), R).
27
3. The transition functions satisfy the cocycle condition, i.e. on the intersection of three
charts Ui ∩ Uj ∩ Uk �= ∅
efij (z) efjk (z) efki (z) = 1 ,
Mij Mjk Mki =
2(n+1)
.
(2.68)
Comparing this definition with the one for rigid special geometry one can see the following
differences. Local special geometry shows a projective structure: There are n physical
scalars in the theory but we can describe everything equivalently in terms of the (n + 1)
functions X I . We call these functions coordinates on the big moduli space, and they are
Xi
related to the physical scalars z i by z i = X
0 , i ∈ {1, . . . , n}. Here the projective nature of
the X I is manifest. Multiplying them with an arbitrary holomorphic function ef (z) does
not change the physics as the scalars z i are not touched by such a transformation. Another
important point to mention is that the prepotential is now more restricted than in the case
of rigid special geometry. Not only has it to be holomorphic but also it is required that
the prepotential is homogeneous of degree two, i.e. F (λX) = λ2 F (X). This implies
FI = FIJ X J ,
FIJK X K = 0
(2.69)
2
∂ F (X)
with FI = ∂F∂X(X)
I , FIJ = ∂X I ∂X J etc.
The problem with having a definition based on a prepotential is that this function is
in general not invariant under symplectic transformations [57]. Therefore one would like
to define special geometry independently of a prepotential. There are even supergravity
actions for which there does not exist a prepotential. An example is given in [59]. This is
not a contradiction to the definition of special manifolds given above. In [57]the authors
state that the geometry of the scalars can always be described in terms of a prepotential.
But this does not exclude the possibility that there are supergravity actions which cannot
be obtained using a prepotential. It is still possible that even when one starts with a theory
with prepotential the symplectically transformed theory cannot be written down in terms
of a prepotential.
For the applications we developed in N = 2, D = 4 we always work with models based
on a prepotential, so we do not need to bother about these kind of subtleties. However,
let us display an alternative definition of special geometry (cf. [57]):
Definition. An n-dimensional Kähler-Hodge manifold M is special Kähler if it has the
following two properties:
1. There exists a holomorphic Sp(2(n + 1), R) vector bundle H over M and a holomorphic section v(z) of L ⊗ H, such that the Kähler form is given by
K=−
i ¯
∂ ∂ log(i�v̄, v�) .
2π
(2.70)
L denotes a holomorphic line bundle over M with transition functions given by U (1)
transformations. The first Chern class of L has to be equal to the cohomology class
of the Kähler form.
28
2. The section v(z) satisfies
�v, ∂α v� = 0 .
(2.71)
It is possible to formulate similar
� I �expressions as in the above definition using a different
X
section V = eK/2 v with V =
. With
FI
1
Uα := Dα V = ∂α V + (∂α K)V ,
2
1
Dᾱ V = ∂ᾱ V − (∂ᾱ K)V
2
(2.72)
and analogously for the complex conjugates one can derive the following constraints [57]:
�V, V̄ �
Dᾱ V
�V, Uα �
�Uα , Uβ �
=
=
=
=
i,
0,
0,
0.
The first one of these equations is equivalent to
�
�
i X̄ I FI − F̄I X I = 1 .
(2.73)
(2.74)
We will use this equation later in the section on N = 2, D = 4 gauged supergravity to
put a constraint on the scalar kinetic term in the big moduli space. This constraint will
be very important because of the projective nature of special Kähler manifolds. We can
Xi
either describe everything in terms of the n physical scalars z i = X
0 or in terms of the
(n + 1) functions X I . But then we need a constraint to get rid of the additional degree of
freedom. Further identities of special geometry which we will use in chapter 5 are presented
in appendix A.
Since later we will not only work in four dimensions but also in five, let us have a look
at the scalar geometry in five dimensions.
2.4
The scalar geometry for N = 2 supergravity in five
dimensions
Also in five dimensions the scalar manifold is equipped with a typical structure. In contrast to the four-dimensional case the manifolds for five-dimensional supergravity are real
manifolds instead of complex ones. These manifolds are called very special (real) manifolds. When one performs a dimensional reduction to four dimensions one is lead to special
Kähler manifolds. Of course, this does not imply that every special Kähler manifold can
be obtained by dimensional reduction from a very special manifold. The map that takes a
very special manifold to a special one is called the r-map (cf. [60]).
The geometry of the scalar manifold in five-dimensional supergravity with Abelian
vector multiplets was worked out in [61]. We will present the essential results and display
some identities for very special geometry which we use in chapter 6.
2.4 The scalar geometry for N = 2 supergravity in five dimensions
29
First of all let us have a look at the field content we are dealing with in five dimensions.
In five-dimensional supergravity one has the gravity multiplet and three kinds of matter
multiplets, namely the vector, tensor and hypermultiplets ( cf. e.g. [62]). Each of the
latter contain scalars but we will focus only on the ones in the vector multiplet since only
these will be necessary for the applications in chapter 6.
The gravity multiplet contains (just as in four dimensions) the graviton, two gravitini
and the graviphoton. Each vector mulitplet contains a vector field, one SU (2) doublet of
spin- 12 fermions and one real scalar field [62]. One can see here that the number nS of
scalars {φi }i=1...,nS is related to the number n of vector fields by
nS = n − 1
(2.75)
since there is one more vector field than scalars, namely the graviphoton.
In [61] it is shown that the (n − 1)-dimensional scalar manifold in this case can be
embedded into an n-dimensional Riemannian space in a particular way. More precisely,
the scalar manifold is a hypersurface with vanishing second fundamental form and the
embedding is given by the cubic polynomial
1
CABC X A X B X C = v .
6
(2.76)
Here v is a constant, the X A , A = 1, . . . , n are real coordinates on the n-dimensional
Riemannian manifold and CABC are constants that have to be chosen in a way such that
CABC is symmetric in its indices. Often the constant v is taken to be equal to one. But for
some applications it is necessary to choose a different value for v as you can see in section
6.2 and appendix D where we used v = 12 .
The physical scalars φi can now be interpreted as coordinates on the (n−1)-dimensional
hypersurface determined by (2.76). Later we will not work with the physical scalars but
with the X A instead – again we will work in big moduli space. For the purpose of using
X A instead of φi we have to relate both kinds of scalar fields. To do so we need to take
into account the form of the bosonic part of the supergravity action with Chern-Simons
term in five dimensions. For Abelian vector fields the Lagrangian density for this action is
given by (cf. e.g. [63, 64])
�
�
√
1
1
i M j
A
B MN
A
B
C KLM N P
L = −g R−gij ∂M φ ∂ φ − GAB FM N F
−
CABC FKL
FM
(2.77)
N AP �
2
24v
A
Here M, N, K, L, P ∈ {t, r, x, y, z}, A, B, C ∈ {1, . . . , n} and i, j ∈ {1, . . . , n − 1}. FM
N
are the field strengths associated to the vector fields.
Now we want to relate the various terms in the action to very special geometry. The
identities displayed below are taken from [63].
If one defines
1
XA = CABC X B X C ,
(2.78)
6
�
�
1
1
9 XA XB
GAB =
− CABC X C +
,
(2.79)
v
2
2 v
30
then one gets the following relations:
XA X A = v
(2.80)
and with GAB the inverse of GAB
XA =
2v
GAB X B ,
3
XA =
3 AB
G XB .
2v
(2.81)
Here GAB is the matrix in the supergravity action appearing in the vector kinetic terms.
Here one can see the interplay of the big moduli scalars X A and the vector fields.
v = const implies ∂i v = 0 and therefore
(∂i XA )X A = 0 .
(2.82)
From the equations (2.78) and (2.79) it follows that
GAB ∂i X B = −
3
∂XA .
2v
(2.83)
The metric on the scalar manifold is given in terms of the X A through
gij = GAB ∂i X A ∂j X B
(2.84)
so that the scalar kinetic term becomes
gij ∂M φi ∂ M φj = GAB ∂M X A ∂ M X B .
(2.85)
In [63] it is also shown that
2
g ij ∂i X A ∂j X B = GAB − X A X B .
3
(2.86)
These are the tools one needs to deal with very special geometry.
2.5
Gauged supergravity in four and five dimensions
In chapters 5 and 6 we will not work with pure supergravity but with a more complicated
version called gauged supergravity instead. Gauged supergravities allow for a broader class
of black solutions and therefore these theories of gravity are of particular interest.
The symmetries of a generic supergravity theory are organized by its global symmetry
group G. For theories with maximal or half-maximal number of supersymmetries the
symmetry groups are uniquely determined in the various dimensions. A list of them can
be found e.g. in [65]. Maximal supergravities in various dimensions have for instance
been studied in [66, 67, 68, 69, 70]. In the case of N = 2 supergravity the situation is
different. Here the symmetry group is less constrained and there are several possibilities
depending on the specific model under consideration. Gauging the supergravity means
2.5 Gauged supergravity in four and five dimensions
31
that a subgroup G0 of the symmetry group G is chosen and promoted to a local symmetry.
This makes it difficult to explain how the gauging works for N = 2 since we do not have
a fixed symmetry group G to work with. The examples we are interested in will be U (1)
gauged supergravities. Here a U (1)k group serves as gauge group with k ≤ n + 1 and n + 1
the number of Abelian vector fields4 . This is always possible due to the Abelian gauge
symmetry U (1)(n+1) of the vector fields. In the general case the subgroup G0 can be chosen
via the embedding tensor formalism as is explained in [65]. This works as follows. One
selects a subset of elements in the Lie algebra g of G that generate a subalgebra. These
generators are related to the generators of G by
t̃I = ΘαI tα .
(2.87)
Here ΘαM is a constant tensor called the embedding tensor and tα are the generators of the
symmetry group G. One can see here that the embedding tensor is a tool to construct a
subalgebra of the Lie Algebra g which is the Lie algebra of the subgroup G0 that will be
promoted to a local symmetry. The embedding tensor has to fulfill a set of constraints to
ensure local gauge invariance (cf. e.g. [65]). With the help of the generators t̃I one can
perform the gauging in the standard way by defining a covariant derivative ( cf. [65])
Dµ = ∂µ − gAIµ t̃I .
(2.88)
Here g is the coupling constant and AIµ the vector fields that participate in the gauging. In
the cases we consider in the chapters 5 and 6 the covariant derivative would only apply to
the scalar fields in the hypermultiplets which we set to zero anyway. Therefore we do not
need to deal with gauge covariant derivatives. In the general non-Abelian cases the field
strengths of the vector fields have to be modified [65]. But for U (1) gauged supergravity
the field strengths do not change because of the commutativity of the group multiplication.
Now there are two things one has to take care of when constructing the Lagrangian
for gauged supergravity: One has to make sure that it is gauge invariant and also that
it is supersymmetric. To ensure both required properties one adds at first terms to the
Lagrangian to make it gauge invariant and then applies the Noether procedure to restore
supersymmetry [65]. This results in a scalar potential that appears at the order g 2 . For
the Lagrangians we will later use this scalar potential is the only modification we see
compared to the ungauged case. The field strengths are left unchanged because we are
looking at U (1)’s and also the covariant derivative is ”invisible“ since we do not consider
hypermultiplet scalars.
In the scalar potential additional parameters appear which are called fluxes. Although
in our calculations we will treat them just as a given set of parameters in four respectively
five dimensions, one could ask where these parameters come from. Indeed there is a
description of them in higher dimensions. In ten-dimensional type II superstring theory
the fluxes are defined as integrals of field strength over appropriate cycles. For instance,
the field strength of the NS B-field
H = dB
(2.89)
4
Here we stick to our notation for the four-dimensional case.
32
in a Calabi-Yau compactification gives rise to electric and magnetic fluxes through [71]
�
�
1
1
K
H
=
h
,
H = hK .
(2.90)
(2π)2 α� AK
(2π)2 α� B K
Here (AK , B K ) is a basis of 3-cycles and hK , hK are the magnetic and electric fluxes.
Turning on fluxes corresponds to gauging some shift symmetries in the hypermultiplet
sector. In [72] it was shown that the scalar potential of gauged supergravity agrees with
the one obtained from compactifying type IIB supergravity on a Calabi-Yau manifold with
fluxes. The author considered electric RR and NS fluxes and he showed that it is possible
to express the Killing vectors for certain shift symmetries of the hypermultiplet scalars in
terms of the electric fluxes. The covariant derivative defined in (2.88) can be expressed in
terms of the Killing vectors according to (cf. [71])
Dµ = ∂µ − kI AIµ
(2.91)
with the Killing vector kI . From this example one can see the general mechanism. The
fluxes coming from higher-dimensional field strengths are related to isometries in the hypermultiplet sector. These isometries are the shift symmetries we have already mentioned.
Depending on which fluxes are switched on different shift symmetries are selected. Since
the isometries of the hypermultiplet sector (like shift symmetries) are part of the symmetry
group of the theory, one can use them to construct the subgroup of the symmetry group
that will be gauged. The fluxes will then enter in the scalar potential of gauged supergravity.
In chapters 5 and 6 we will use Lagrangians from N = 2 U (1) gauged supergravity to construct black brane solutions. These solutions are interesting since they allow for asymptotic
AdS solutions which is important for applications within the AdS/CFT correspondence.
The next chapter will be devoted to introducing this correspondence.
Chapter 3
The AdS/CFT Correspondence
Circle Limit IV, Maurits C. Escher
Several artworks of the Dutch artist Maurits C. Escher1 deal with the artistical arrangement
of a hyperbolic plane which can be identified with a disk. One of these is the xylograph
”Circle Limit IV”.
As you can see, the angels and devils become smaller and smaller the closer they get
to the boundary. The size of the figures is a measure for their distance from the centre of
the hyperbolic plane. From the geometrical point of view this means that the boundary is
infinitely far away from the centre and no angel or devil could ever reach it.
Now you will certainly ask what this could have to do with physics. In this section we
will review the AdS/CFT correspondence and as you will see shortly, the ”AdS” part of
that correspondence indeed involves hyperbolic spaces.
1
c
All M.C. Escher works �2013
The M.C. Escher Company - the Netherlands. All rights reserved. Used
by permission. www.mcescher.com
34
3. The AdS/CFT Correspondence
The AdS/CFT correspondence was proposed in 1997 by Maldacena [5].His famous
conjecture states that IIB string theories on Anti-de Sitter spacetimes are dual to conformal
field theories living on the boundary of these Anti-de Sitter spaces [5]. This means that
there are two different theories that describe the same physics. It is just like in Escher’s
xylograph: Do you see the angels or the devils? Both describe the same arrangement of
black and white.
To understand the AdS/CFT conjecture we need to introduce its basic ingredients.
These are the Anti-de Sitter spaces we have already mentioned as well as the conformal
field theory under consideration, namely N = 4 super-Yang-Mills theory. So let us start
with the field theoretic side of AdS/CFT.
3.1
The field theory side of the duality
We will describe the main aspects of N = 4 super-Yang-Mills theory in four dimensions.
Here we will mostly rely on [73]. To make it more precise, to understand AdS/CFT
we need to deal with N = 4 SU (N ) super-Yang-Mills theory. As we have seen before
theories with N = 4 supersymmetry contain four spinorial supercharges with commutation
relations given in (2.35). These supercharges can be rotated into each other by SU (4)
transformations. Any theory dual to an N = 4 supersymmetric theory should better also
display this kind of symmetry, and later we will see that a SU (4) symmetry appears indeed
in the gravitational dual to N = 4 SYM. The field content of N = 4 SYM consists only of
a gauge multiplet. This contains four left Weyl fermions λaα , a = 1, . . . , 4, six real scalars
X i , i = 1, . . . , 6 and a gauge field Aµ . Since we are considering a SU (N ) theory, Aµ is a
SU (N ) gauge connection. For AdS/CFT it will be of particular interest to examine the
large N behavior of the SU (N ) super-Yang-Mills theory.
The Lagrangian of N = 4 SYM takes the form (cf. [74])
�
L = tr −
4
6
�
�
1
θI
µν
µν
a µ
F
F
+
F
F̃
−
i
λ̄
σ̄
D
λ
−
Dµ X i Dµ X i
µν
µν
µ a
2gY2 M
8π 2
a=1
i=1
�
�
�
gY2 M � i j 2
ab
i
a
i
b
[X , X ] .
+
gY M Ci λa [X , λb ] +
gY M C̄i ab λ̄ [X , λ̄ ] +
2 i,j
a,b,i
a,b,i
(3.1)
Here Ciab and Ci ab are the structure constants of SU (4). We will now review the symmetries
of this Lagrangian. Depending on whether �X i � = 0 for all i or not there are two different
phases for N = 4 SYM. If �X i � = 0 for all i the theory is said to be in the superconformal
phase, otherwise it is in the Coulomb phase. In the superconformal phase none of the
symmetries is broken. So let us specify what these symmetries are.
First of all the SYM Lagrangian is invariant under translations and Lorentz transformations. In addition, N = 4 SYM has conformal symmetry. A transformation is conformal if
it preserves the angle between two vectors. So every isometry (like translations and Lorentz
3.2 The gravitational side of the duality
35
transformations) is of course a conformal transformation but there are further conformal
transformations which are no isometries. Conformal transformations of this type are the
dilations
xµ �→ λxµ , λ ∈ R
(3.2)
and the so-called special conformal transformations which can be written in the form (cf.
e.g. [75])
xµ + α µ x2
xµ �→
,α
� ∈ R4 .
(3.3)
1 + 2��
α, �x� + α2 x2
The conformal transformations and the supersymmetry transformations can be combined to form the symmetry group of N = 4 SYM. This symmetry group is called the
supergroup SU (2, 2|4). It has the following constituents (cf. [73]):
• The conformal transformations form the conformal group SO(2, 4) ∼ SU (2, 2). Here
”∼” means that SU (2, 2) is the universal cover of SO(2, 4). This difference will not
be important for our purposes, so we will treat both groups as if they were identical.
The generators of the Lie algebra of SO(2, 4) are the translations, Lorentz transformations, dilations and special conformal transformations.
• The SU (4) symmetry which rotates the supercharges Qaα into each other. This is
called the R-symmetry.
• The supersymmetries generated by the supercharges Qaα and their conjugates.
• Furthermore, there are additional supersymmetries Sα a and their conjugates. These
are called conformal supersymmetries. Their existence is due to the fact that the
supercharges Qaα do not commute with the special conformal transformations and
so their commutator has to be an additional symmetry. These are the conformal
supersymmetries.
These generators form a Lie algebra and the group associated to this Lie algebra is called
SU (2, 2|4).
Now we can proceed looking at the gravitational part of the AdS/CFT conjecture.
3.2
The gravitational side of the duality
At first we will define Anti-de Sitter spaces of arbitrary dimension and give some explanation of what is called the boundary of this space. We will also review the symmetries of
AdS spaces and show how these spaces enter in the AdS/CFT correspondence, namely in
a low energy limit of some closed string theories.
AdS spacetimes arise as solutions to the Einstein-Hilbert action with cosmological constant (cf. e.g. [75]):
�
�
1
S = 2 dD x |g|(R − Λ)
(3.4)
2κ
36
Here Minkowskian signature is used. κ is related to the D-dimensional Newton’s constant
GD through κ2 = 8πGD . R denotes the Ricci scalar and Λ = (D−1)(D−2)
is the cosmological
L2
constant. The meaning of L will become clear momentarily. For AdS spaces we have
Λ < 0. AdS spaces are spacetimes with constant negative scalar curvature. The D = d + 1
dimensional solution can be embedded as a submanifold into R2,d . This is a space with
indefinite inner product given by
(x, y) = �x, diag(−1, −1, 1, . . . , 1, 1) y�
(3.5)
and � , � the standard scalar product on Rd+2 . For y = (y0 , y1 , . . . , yd , yd+1 ) ∈ R2,d the
”length squared” is determined by the indefinite inner product, y 2 = (y, y), and we can
describe AdSd+1 as the quadric (cf. e.g. [73, 75])
y 2 = −R2 = const.
(3.6)
for a real number R. This is a (d + 1)-dimensional surface with negative curvature in R2,d .
From this embedding we can see a fact that will be important in understanding Maldacena’s conjecture. It is well known from Mathematics that the group of rotations in R2,d
is SO(2, d). Since rotations preserve the inner product, it is clear that the quadric (3.6) is
invariant under the transformation �y �→ Λ�y for Λ ∈ SO(2, d). But this means that every
SO(2, d) transformation is an isometry of AdSd+1 . One can see that in particular AdS5
has a SO(2, 4) rotational symmetry. We met SO(2, 4) already as the conformal group in
four dimensional space when we discussed the symmetries of N = 4 SYM, so here is the
first indication of a possible duality.
But let us come back to the geometry of AdSd+1 . The following discussion is based on
[76]. The surface (3.6) is a connected hyperboloid in the (d + 2)-dimensional ambient space
R2,d (see figure 3.1). One can then introduce coordinates (t, ρ, Ωi ) on the hyperboloid such
that
y0 = R cosh(ρ) cos(τ ) ,
y1 = R cosh(ρ) sin(τ ) ,
yi = R sinh(ρ) Ωi , i ∈ {2, . . . , d + 2} ,
�
Ω2i = 1 .
(3.7)
i
When taking ρ ∈ R+
0 and τ ∈ [0, 2π) the entire hyperboloid is covered once. The “time”
coordinate τ is periodic and therefore AdSd+1 has closed timelike curves. One can avoid
this problem by going to the universal cover instead since in this space the circle in the
τ direction is unwrapped. Often also the universal cover of AdSd+1 is itself called AdSd+1
and in our applications in the later chapters we also work with the universal cover.
There are two further important aspects of AdSd+1 . Firstly, one can change to imaginary
time. The Euclidean version of AdSd+1 is then given by the hyperbola
− (y 0 )2 +
d+1
�
i=1
(y i )2 = −R2 .
(3.8)
37
Figure 3.1: AdS space in Minkowski signature (on the left) and Euclidean signature.
As you can see in figure 3.1 this is now a disconnected hyperboloid. AdSd+1 is considered
to be only one of the two connected leaves of the hyperboloid when one wants to work in
a connected space.
Secondly, one can identify AdSd+1 with the upper halfplane
Hn+1 = {(z0 , �z )|z0 ∈ R+ , �z ∈ Rn } .
(3.9)
This can be done by the coordinate transformation [73]
y0 + y1 =
1
,
z0
zi = z0 yi , i = 2, . . . , d + 1 .
(3.10)
The metric on Hd+1 is then given by
ds2 =
1
(dz 2 + d�z 2 ) .
z02 0
(3.11)
By adding a point at infinity the upper halfplane Hd+1 can be compactified. This procedure gives the unit ball Bd+1 as the compactification of Euclidean AdSd+1 . Often both
descriptions are used equivalently and no distinction is made between AdSd+1 and its
compactification.
Now that we got to know AdSd+1 we are interested in the structure of its boundary.
To examine the boundary of AdSd+1 one can introduce ”light cone coordinates” [75] 2
u = y0 + iy1 ,
2
We use a different sign convention than [75].
v = y0 − iy1 .
(3.12)
38
In these coordinates AdSd+1 is given by
− uv + �y 2 = −R2
(3.13)
with �y = (y2 , . . . , yd+1 ). The ”boundary” of AdSd+1 are the points lying at infinity. For
the quadric (3.13) these points are given by
uv − �y 2 = 0 .
(3.14)
This can be seen in the following way. Going to infinity means that we consider points of
the form λ(u, v, �y ) with large |λ|. For such points in AdSd+1 (3.13) takes the form
− λ2 uv + λ2 �y 2 = −R2 .
(3.15)
Dividing this equation by λ2 �= 0 and taking the limit λ → ∞ yields (3.14). One can see
immediately that for a point (u, v, �y ) satisfying (3.14) also λ(u, v, �y ) fulfills (3.14) for any
λ ∈ R. This means that the boundary of AdSd+1 is the set of equivalence classes of points
satisfying (3.14).
This set of equivalence classes itself is furnished with a projective structure. Therefore
the boundary of AdSd+1 is automatically compactified. For the Euclidean version it has the
topology of the sphere S d (cf. e.g. [73]) whereas in the Minkowskian case the boundary
is the conformal compactification of the d-dimensional Minkowski space (cf. e.g. [76]).
But let us come back to the Euclidean case and examine the properties of the boundary
of AdSd+1 a bit in more detail. One can consider the points with v = 0 as the points at
infinity. We are looking now at the boundary itself, i.e. “infinity“ corresponds now to the
points that constitute the projective structure of the boundary and not of the entire AdS
space.
For boundary points with v �= 0 we can assume v = 1 because of the scaling invariance
and then rewrite (3.14) as
u = �y 2 .
(3.16)
Since this equation defines a d-dimensional surface, we can see that the boundary of AdSd+1
has indeed dimension d as desired. These arguments are presented in more detail in [75].
There it is also shown that the group of rotations of AdSd+1 acts as the conformal group
on the boundary of AdSd+1 . In [75] this is discussed for the Euclidean case. Then the
group of rotations is SO(1, d + 1) and it acts on the boundary of AdSd+1 as
Λ[y] := [Λy] .
(3.17)
[y] = [(u, v, �y )] = {(u� , v � �y � )|∃λ ∈ R with (u� , v � , �y � ) = λ(u, v, �y )} ,
(3.18)
Here Λ ∈ SO(1, d + 1) and
i.e. [y] is the equivalence class of the boundary point y = (u, v, �y ). In [75] it is stated
that a transformation Λ ∈ SO(1, d + 1) infinitesimally close to the identity can be written
as the composition of infinitesimal conformal transformations. This shows that the group
39
Figure 3.2: Open strings attached to branes.
of rotations in Euclidean space acts as the conformal group on the boundary of AdSd+1 .
This result can be transferred to the Minkowski case and we can conclude that the group
SO(2, d) acts as the conformal group on the boundary of Minkowskian AdSd+1 . From this
point of view it makes sense to say that the conformal field theory N = 4 SYM lives on
the boundary of AdS5 because both are equipped with SO(2, 4) conformal symmetry.
But how do AdS spaces arise from string theory? They appear in the low energy limit
of Dp-brane solutions to 10D string theory! Generally speaking a Dp-brane is just an
object on which open strings can end (see figure 3.2). The ”D” stands for ”Dirichlet” since
the coordinates of the strings attached to the brane fulfill Dirichlet boundary conditions in
the directions transverse to the brane. A Dp-brane is a p-dimensional brane (cf. e.g. [77]).
Dp-brane metrics are solutions in type II string theory. For the basic understanding of the
AdS/CFT correspondence solutions with N coincident D3 branes are of particular interest,
so we will restrict ourselves to the discussion of this kind of branes. Such a solution is then
given in terms of the metric (cf. e.g. [73])
2
ds =
�
L4
1+ 4
r
�− 12
�
L4
ηij dx dx + 1 + 4
r
i
j
� 12
(dr2 + r2 dΩ25 ) .
(3.19)
Here ηij , i, j ∈ {0, 1, 2, 3} denotes the metric on four dimensional Minkowski space and
dΩ25 is the usual line element on the unit sphere in R6 . The radius L is related to the
number of coincident branes by
L4 = 4πgS N (α� )2 .
(3.20)
Furthermore, D3-brane solutions in string theory have constant axion and dilaton. Both
are scalar fields in string theory. The dilaton determines the string coupling constant by
gS = eφ with φ the expectation value of the dilaton field (cf. e.g. [73]). The axion is a
40
Goldstone boson originating from spontaneously broken Peccei-Quinn symmetry (cf. e.g.
[78])3 .
However, in supergravity Dp-brane solutions are known to be 12 -BPS states, i.e. they
preserve half of the supersymmetries (cf. [73]).
But let us now look at the asymptotic behavior of the D3-brane metric (3.19) for r → 0
and r → ∞. One can see immediately that for r → ∞ the metric becomes flat. The socalled near-horizon limit r → 0 can be examined as follows (cf. e.g. [73]). Defining the
2
new variable u = Lr and taking u → ∞ one can see that asymptotically the line element
(3.19) becomes
�
�
1
1 2
2
2
i
j
2
ds = L
ηij dx dx + 2 du + dΩ5 .
(3.21)
u2
u
This of the form AdS5 × S 5 with AdS radius and radius of the sphere each given by L.
One can see that the horizon region is infinitely far away from any observer, since it lies
at the end of an infinite throat as it was also the case for the horizon of an extremal black
hole4 .
Now that we have learned about N = 4 SU (N ) SYM and AdS spaces we are ready to
state the AdS/CFT correspondence.
3.3
The AdS/CFT correspondence
The AdS/CFT correspondence states a duality between type IIB string theory on AdS5 ×S 5
and the conformal field theory N = 4 SYM in four dimensions which is said to live on the
boundary of AdS5 . This can be motivated as follows. In the presentation of the basic ideas
we stick closely to [81]. As of now the number N of coincident branes respectively the
dimension of the fundamental representation of the SU (N ) gauge group is considered to
be large. The principal idea of AdS/CFT is then to increase gS N from very small values
to very large ones and look at low energy excitations. Then one compares the decoupled
theories one gets for small and for large gS N .
For small values of gS N we can neglect gravitational effects of the branes and so we
can treat them just if they were in flat space. One can see from (3.20) that for gS N → 0
L goes to zero as well and therefore the metric (3.19) becomes flat. The only effect of the
D3-branes is that we have to determine appropriate boundary conditions for the strings
ending on them. We will discuss the case gS N small in more detail below.
In contrast, for large values of gS N gravitational effects are important and we need
to deal with the complicated geometry of the D3-branes that curve the spacetime in a
non-trivial way.
As we have seen before the near horizon geometry is described by AdS5 × S 5 . For
understanding AdS/CFT we need to consider on the one hand low energy excitations of
3
This symmetry was postulated to solve the strong CP problem [79]. The axion is seen as a possible
dark matter candidate. The role of axions in string theory is discussed for instance in [80].
4
The calculation here works analogous to (2.17).
3.3 The AdS/CFT correspondence
41
closed strings near the horizon and on the other hand such modes that propagate in the
asymptotic flat region far away from the horizon.
An observer in asymptotically flat space perceives a very low energy for modes with
finite energy coming from the horizon region since these modes are redshifted. The closer
such modes are brought to the horizon the harder it becomes for them to escape from
the throat. But this means that an observer at asymptotic infinity just cannot see those
modes at all, they are trapped in the throat (cf. e.g. [76]). Low energy modes in the
asymptotically flat region correspond to modes with large wavelength compared to the size
of the horizon. These excitations are not affected by the throat geometry, they do not see
the mouth of the throat. More precicely, one can compute the absorption cross section
for low energy excitations. This quantity goes to zero when the energy is lowered (cf. e.g.
[76]). Such computations have been done in [82, 83].
All in all we have two systems of low energy modes in type IIB string theory that
decouple from each other. One is a system of low energy closed strings propagating in flat
ten dimensional Minkowski space. The second system are closed strings in the curved near
horizon region with geometry AdS5 × S 5 .
So what happens for small values of gS N ? Since G10 ∼ gS2 (α� )4 (cf. [81]), G10 goes
to zero in the case gS N << 1, which implies gS << 1 for N large. But G10 governs
on the one hand the interactions of closed strings propagating in spacetime and on the
other hand the interactions of spacetime fields and fields on the branes. So when the
coupling goes to zero we get a system of decoupled closed strings in 10D Minkowski space
and secondly the fields on the branes. Here we consider the N D3-branes as surfaces on
which open strings can end and do not take into account any gravitational effects from
the branes. Low energy excitations of Dp-branes are always governed by supersymmetric
non-Abelian gauge theories (cf. e.g. [77]), and the gauge theory describing the low energy
modes of strings ending on D3-branes is N = 4 SU(N) Yang-Mills theory. Also in the case
gS N << 1 we get two decoupled systems. Again one are low energy closed strings in flat
ten dimensional space and the second one is the SYM theory describing the dynamics on
the branes.
Comparing the situation for small values of gS N and for large ones, we can see that we
always end up with two decoupled systems one of which are closed strings in flat space.
The idea of AdS/CFT is now that the respective second system describes the same physics.
A priori in SU (N ) SYM it is not necessary to restrict to small values of gS N . Similarly,
type IIB string on AdS5 × S 5 theory is also a good theory for small values of gS N and not
only for large ones. So if both theories do not need a particular range of gS N , why should
we not assume that they are both valid for all values of gS N and that the two decoupled
systems one gets in both cases are equivalent?
The AdS/CFT correspondence conjectures exactly this. N = 4 SU (N ) Yang-Mills and
type IIB superstring theory on AdS5 × S 5 are dual to each other. A theory of gravity in a
very non-trivial geometry and a conformal supersymmetric field theory describe the same
physics!
At this point it is necessary to mention that at present there has been no rigorous proof
of the AdS/CFT conjecture. Today AdS/CFT has the status of a conjecture. Nonetheless,
42
all calculations done within this framework indicate that the AdS/CFT correspondence
holds.
What makes AdS/CFT also appealing is that the idea of holography is present in
it. As already mentioned the field theory is often said to live on the boundary of AdS.
When the field theory on the boundary and the gravitational theory in the bulk describe
the same physics, all relevant information is already encoded in the boundary which has
one dimension less than the bulk. This is analogous to a hologram in which all threedimensional data is incorporated in a two-dimensional surface.
It is possible to state the AdS/CFT correspondence in different versions (cf. e.g. [73]).
The strongest one postulates an equivalence of N = 4 SYM theory in four dimensions and
the full quantum type IIB string theory on AdS5 × S 5 for every value of gS N with the
identifications gS = gY2 M and as mentioned earlier L4 = 4πgS N (α� )2 . N is also related to
the 5-form flux through the sphere S 5 (cf. e.g. [73]). The problem is that at present there
is no tractable string theory quantization on a curved manifold like AdS5 × S 5 so that we
need to turn to versions of AdS/CFT that are accessible in an easier way.
One possibility is to look at the t’Hooft limit. Here the t’Hooft coupling
λ = gY2 M N = gS N
(3.22)
is kept fixed while letting N → ∞. The t’Hooft coupling controls the field theory amplitudes in the large N limit. This can be seen in the following way (cf. e.g. [81]). Looking
at the open strings propagating on the N branes, splitting and rejoining again, we can describe their interactions in a diagrammatic way. The amplitude for a string having its ends
on the i-th and j-th brane and apart from that doing nothing at all is simply a constant
without any dependence on N . The situation is different when the string splits into two
strings and rejoins again. Now there are two interaction vertices either of which produces
a factor of gY M in the amplitude (see picture a) in fig. 3.3). Moreover, we need to sum
over all possibilities for the brane on which the string could split. This results in a factor
of N for the N possible branes. In summa, we get for the amplitude
A1 = c1 gY2 M N ,
(3.23)
with c1 a constant. The same method applies when we are considering a string splitting
n times. For each splitting we get an additional factor of gY2 M N , and two interaction
points as well as one boundary are added to the diagram (see picture b) in fig. 3.3). To
summarize, the full amplitude for these kinds of diagrams is given through
A=
∞
�
n=0
cn (gY2 M N )n =
∞
�
c n λn .
(3.24)
n=0
Up to here we only looked at interactions where new boundaries are created, but it is also
possible to add strips to the diagrams that decrease the number of boundaries (see picture
c) in fig. 3.3). Such diagrams cannot be drawn on a sheet of paper or a blackboard since
they are not planar in contrast to the ones we considered before. The non-planar diagrams
3.3 The AdS/CFT correspondence
43
a)
b)
c)
Figure 3.3: Planar and non-planar diagrams for string interactions.
contribute to the full amplitude by a factor of
number of boundaries.
The full amplitude is then given by
A=
∞
�
n=0
cn λn + f2 (λ)
2
gY
M
N
=
λ
N2
for each strip that decreases the
1
1
+ f4 (λ) 4 + . . . .
2
N
N
(3.25)
At the moment it is not necessary to determine the functions fi (λ) explicitly. From (3.25)
one can see immediately that for N → ∞ while λ is fixed only the summands with positive power of N contribute while the others disappear in the large N limit. Further,
the amplitude is determined in terms of the t’Hooft coupling rather than the Yang-Mills
coupling.
So the t’Hooft limit is well-defined on the field theoretic side. On the gravity side (3.25)
corresponds to a string loop expansion. Here gS = Nλ → 0 for fixed λ and N → ∞, so we
reach a weak coupling regime. Therefore we may treat the type IIB theory perturbatively
in gS .
Another important limit is the Maldacena limit. Here we let in addition to N → ∞
also λ → ∞. This corresponds to strong coupling on the field theory side. On the other
hand, on the AdS side we have (cf. (3.20))
L2 √
= 4πλ .
α�
(3.26)
From this one can see that λ → ∞ implies α� → 0, and we end up with classical type IIB
supergravity. From here we can see that AdS/CFT provides a tool for treating strongly
coupled field theories. Instead of performing calculations in a hardly tractable strongly
coupled regime in SYM we could do the computation in supergravity!
44
Not only that AdS/CFT supports a duality between theories for open and closed strings
but also between strongly and weakly coupled theories. Before we come to a more rigorous
formulation of AdS/CFT, let us look at a further hint of the duality, namely the symmetries
of both theories (cf. e.g.[73, 75]).
We have already reviewed the symmetry group of N = 4 SU(N) SYM and found it to
be the supergroup SU (2, 2|4) which includes the conformal group SU (2, 2) ∼ SO(2, 4) and
the R-symmetry group SU (4) ∼ SO(6) as subgroups.
Both groups appear also on the gravity side. As mentioned before, the elements of
SO(2, 4) describe isometries of AdS5 . Furthermore, the isometries of the sphere S 5 are
simply the elements of SO(6) as those are rotations in R6 .
3.4
The AdS/CFT dictionary
Having described the AdS/CFT correspondence in a more “intuitive” way we can now
proceed giving a more rigorous concept of AdS/CFT. For this purpose we need to relate
gauge invariant operators in N = 4 SYM on the one hand and fields in IIB theory on
the other hand. This correspondence is given through the field-operator map. We follow
mainly [73],[84] and [85]. The ideas were first presented in [84, 86].
Since the N = 4 SYM is interpreted to live on the boundary of AdS5 we actually want
to define boundary values for the bulk fields. We will show the concept for the case of a
scalar field. On AdS5 × S 5 all scalar fields can be decomposed into Kaluza-Klein towers
on S 5 , i.e. they can be expanded in spherical harmonics Y∆ (cf. also the article by J.
Erdmenger in [87]):
∞
�
ϕ(x̃, �y ) =
ϕ∆ (x̃)Y∆ (�y ) ,
(3.27)
∆=0
with �y coordinates on the 5-sphere and x̃ = (r, �x) coordinates on AdS5 equipped with the
metric
L2
ds2 = 2 (dr2 + d�x2 ) .
(3.28)
r
Note that the line element can be taken with Euclidean or Minkowskian signature. Here
the boundary is located at r = 0.
The dimensional reduction to AdS5 yields a massive wave equation for ϕ∆ :
(�5 + L2 m2∆ )ϕ∆ (x̃) = 0 .
(3.29)
The equation of motion for ϕ∆ and also the relation between m∆ and ∆ depends on what
kind of field one considers. For a scalar field the relation is given by 5
L2 m2∆ = ∆(∆ − 4) .
5
For arbitrary dimensions AdSd+1 m∆ is given through L2 m2∆ = ∆(∆ − d) (cf. e.g. [85]).
(3.30)
3.4 The AdS/CFT dictionary
45
The wave equation (3.29) has two linearly independent solutions with different asymptotic behavior for r → 0:
� � �∆
r
(normalizable)
ϕ∆ (r, �x) ∼ � Lr �4−∆
(3.31)
(non − normalizable)
L
One comment is in order here. It is not always a priori clear which mode is the normalizable
one. Usually, the asymptotic behavior of the normalizable mode is determined by the larger
root of (3.30), but in [88] it is argued that for a certain mass range also the smaller root
can lead to a normalizable mode.
Also the case ∆ = 2 is special since then the two solutions in (3.31) coincide. A second
independent solution is obtained by adding a logarithmic term [89]. Here we will not
address these more specific cases.
We can now proceed and define boundary values using the non-normalizable modes:
(0)
ϕ∆ (�x) := lim ϕ∆ (r, �x)
r→0
� r �∆−4
L
.
(3.32)
Correlation functions in N = 4 SYM and fields on AdS5 × S 5 can now be related as follows.
On the gravity side we have the type IIB action S[ϕ∆ ] for the bulk fields ϕ∆ . This
action can be the type IIB supergravity action or might also include α� corrections. Now
(0)
let Z[ϕ∆ ] denote the (string theory or supergravity) partition function for ϕ∆ equipped
with the correct boundary conditions. In the case the supergravity approximation is valid
and ϕ∆ is a solution to the equations of motion, we can write the partition function in
terms of the on-shell supergravity action (cf. [84])
(0)
Z[ϕ∆ ] = exp(−Son−shell [ϕ∆ ]) .
(3.33)
For Minkowski signature this reads
(0)
Z[ϕ∆ ] = exp(iSon−shell [ϕ∆ ]) .
(3.34)
From now on we will drop the subscript ”on-shell” but you should always keep in mind
that the action has to be evaluated on a solution.
One can apply these formulae also outside the supergravity regime, but then one has
to include e.g. string theory corrections (cf. [84]).
The boundary values appear as sources for the field theory operators O∆ via the identification
�
��
��
(0)
(0)
4
Z[ϕ∆ ] = exp
d x ϕ ∆ O∆
.
(3.35)
∂AdS5
The expression in brackets is to be understood as the path integral over the canonical fields
X i , λa and Aµ of N = 4 SYM weighted by the action. The operators one considers are in
general polynomials in the canonical fields or their derivatives. It can be shown that ∆ is
the conformal dimension of the operator O∆ [84].
46
There is an analogous formula to (3.35) for Minkowskian signature (cf. e.g. [85]),
�
� �
��
(0)
(0)
Z[ϕ∆ ] = exp i
d 4 x ϕ ∆ O∆
.
(3.36)
∂AdS5
The equations (3.33) (respectively (3.34)) and (3.35) (respectiveley (3.36)) lay the foundation of the AdS/CFT correspondence. We can compute field theory correlation functions
using the bulk partition function.
For example, the expectation value of an operator O∆ can be computed (using Minkowski
signature) by
(0)
(0)
δ log Z[ϕ∆ ]
δS[ϕ∆ ]
�O∆ � = −i
=
.
(3.37)
(0)
(0)
δϕ∆
δϕ∆
The last equality is valid if (3.34) applies (cf. e.g. [85]). Usually the action will also include
boundary terms. These can come from the renormalization on the one hand but on the
other hand there may also be finite local counter terms in the action (cf. [90]). These finite
counter terms can always be chosen such that the expectation value takes the particular
simple form we display below. It can be shown that for a scalar field with asymptotic
behavior for r → 0 according to (3.31)
� r �4−∆
� r �∆
(0)
(1)
ϕ∆ =
ϕ∆ +
ϕ∆ + . . .
(3.38)
L
L
the expectation value of the dual operator O∆ is given by
�O∆ � =
2∆ − 4 (1)
ϕ∆ .
L
(3.39)
Indeed, the derivation of (3.39) makes use of boundary terms in the action. We refer to [85]
for the details of the calculation. It is often said that the non-normalizable mode gives the
source of the dual operator and the normalizable mode its expectation value. The latter
is now clear in light of (3.39). This result can be generalized for other kinds of fields than
scalars. In AdS/CFT the expectation value of the dual operator is always given in terms of
the coefficient of the subdominant independent solution in the expansion of the field near
the boundary. The coefficient of the dominant term however gives the source term for the
operator. We will see in subsection 3.6 how this works in a particular example.
3.5
Generalizations and applications of AdS/CFT
So far we have established the AdS/CFT conjecture for AdS5 × S 5 and the field theoretic dual N = 4 SYM. But this is not the only possibility for having field theories with
gravitational duals. One straightforward generalization of the presented ideas is to change
from AdS5 × S 5 to different dimensions for the AdS space. In M-theory AdS4 × S 7 and
AdS7 × S 4 are of interest (cf. e.g.[75]). All the formulas from the previous section generalize to different dimensions in a straightforward way. For this reason we will from now on
3.5 Generalizations and applications of AdS/CFT
47
work in arbitrary dimension of the anti-de Sitter space. It is also possible that the sphere
is replaced by a different kind of space. Such a case was considered e.g. in [91] where it
was argued that there exists a field theory dual to string theory on the space AdS5 × X5
with X5 = (SU (2) × SU (2))/U (1).
Moreover, one could ask whether there exist gravity duals for field theories with much
less symmetry than N = 4 SYM. For example, it would be very nice to have a description
of QCD which is strongly coupled in terms of a weakly coupled theory of gravity.
One way of breaking some symmetry on the field theoretic side is to introduce temperature. A temperature gives rise to an energy scale in a natural way. This does not a priori
mean that a theory at finite temperature cannot be conformal. We will see that there are
theories in which all non-zero temperatures are equivalent and therefore the theory is still
conformal. To break conformal invariance one has to introduce a second energy scale, e.g.
a chemical potential. What then really introduces a scale in the theory is the ratio of the
temperature and the chemical potential. We will come back to that shortly.
So how could one introduce in a first step a temperature on the gravity side? Right –
a good idea might be to look at a black hole in anti-de Sitter space!
As already mentioned AdSd+1 is a solution to the Einstein gravity action with negative
cosmological constant, which we recapitulate here for convenience:
�
�
�
√
1
d(d − 1)
d+1
S = 2 d x −g R +
.
(3.40)
2κ
L2
The following discussion is mainly based on [85]. A black hole solution to the equations
of motion derived from this action is the AdS Schwarzschild solution with line element (in
Minkowski signature)
�
�
L2
dr2
ds2 = 2 −f (r)dt2 +
+ dxi dxi
(3.41)
r
f (r)
with
f (r) = 1 −
�
r
r+
�d
.
(3.42)
At r = r+ the tt-component of the metric vanishes and hence there is a horizon at r = r+ .
For r → 0 instead we rediscover the metric of AdSd+1 . For r → 0 we see that f (r) → 1
and so asymptotically the line element looks like
�
�
dt2 dr2 dxi dxi
2
2
ds = L − 2 + 2 +
(3.43)
r
r
r2
which is AdSd+1 .
The temperature of the AdS Schwarzschild black hole is
T =
d
.
4πr+
(3.44)
48
One method to obtain the temperature of a black hole that is different from the method
involving the surface gravity we mentioned in chapter 2 is to Wick rotate to Euclidean time
and to set some periodicity conditions for the Euclidean imaginary time. We will present
this method briefly.
For the AdS black hole one gets in the case t �→ −iτ
�
�
L2
dr2
2
2
i
i
ds = 2 f (r)dτ +
+ dx dx .
(3.45)
r
f (r)
For the solution to be regular at the horizon r = r+ one has to require periodicity of the
Euclidean time according to6
τ ∼τ+
4π
|f � (r
+ )|
=τ+
4πr+
.
d
(3.46)
The temperature of the black hole is then given by the inverse of the periodicity, i.e. we
get for the AdS Schwarzschild black hole exactly formula (3.44). To show that the black
hole temperature is also the temperature of the dual field theory one can argue as follows
like it was presented e.g. in [85]. The bulk metric can be pulled back to the boundary of
(0)
AdSd+1 and gives there rise to a boundary metric gµν subject to
gµν (r) =
L2 (0)
g + ...
r2 µν
(3.47)
for r going to zero 7 . Comparing (3.45) with (3.47) one can see that
ds2,(0) = dτ 2 + dxi dxi
(3.48)
with τ identified periodically as in (3.46). This metric can be taken as the background
metric for the field theory living on the boundary. From quantum field theory we know
that a field theory at finite temperature corresponds to formally having a Euclidean time
with a periodicity of the inverse temperature (cf. e.g. [92]). But this leads to the same
temperature as the Hawking temperature of the black hole in the bulk. The only parameter
appearing in the expression for the temperature is the horizon radius r+ . By the coordinate
transformation (t, r, xi ) �→ r+ (t, r, xi ) the radius r+ can be removed from the metric. But
d
then r+ also does not enter in the temperature which reads now T = 4π
�= 0. This means
that all non-zero temperatures are equivalent and the dual field theory is still conformal.
This can be seen in the following way. Close to the horizon one can approximate f (r) ≈ f � (r+ )(r − r+ )
�
as f (r+ ) = 0. By introducing the new coordinates ρ2 = |f � (r4 + )| (r −r+ ) and φ = |f (r2 + )| τ we can rewrite (in
6
2
the near-horizon approximation) f (r)dτ 2 + fdr(r) = dρ2 + ρ2 dφ2 . To avoid a conical singularity φ ∼ φ + 2π.
This leads to the periodicity of τ in(3.46).
7
Earlier we showed that the boundary of AdS5 has a conformal structure and the same is of course true
for higher dimensional anti-de Sitter spaces. Therefore the metric at the boundary is only defined up to
an overall rescaling but this does not affect the main argument.
3.5 Generalizations and applications of AdS/CFT
49
Here a natural question arises: How can we implement a theory that distinguishes
between different temperatures? To do so we have to include a Maxwell field in the
gravitational action,
�
�
�
�
�
√
1
d(d − 1)
1
d+1
µν
S = d x −g
R+
− 2 Fµν F
(3.49)
2κ2
L2
4g
where F = dA is the field strength. In chapter 4 we will also include non-Abelian gauge
fields, but let us first show the principle mechanisms in the Abelian case. Again we follow
the discussion in [85].
We require that all relevant quantities depend only on the radial variable r. As was
done in the AdS Schwarzschild case for the metric, we assume that the vector potential
can be pulled back to the boundary, i.e.
Aµ (r) = A(0)
µ + ...
for r → 0 .
(3.50)
(0)
To introduce a scale in the field theory one can add e.g. a chemical potential µ = At .
As we have already seen, to break conformal invariance it is not enough to introduce a
temperature in the theory. We need an additional scale with which the temperature can
be compared. This scale will be given by the chemical potential.
Having included a chemical potential one has to look for solutions to the Einstein and
Maxwell equations with a vector potential of the form
A = At (r)dt .
(3.51)
(0)
Another possibility to introduce an energy scale is to add a magnetic field B = Fxy (see
e.g. [93, 94, 95, 96, 97]), but we will restrict ourselves to the case of adding a chemical
potential since we will do exactly this in the computations in chapter 4.
A solution to Einstein-Maxwell theory with a chemical potential is given by the AdS
Reissner-Nordström black hole with line element
�
�
L2
dr2
2
2
i
i
ds = 2 −f (r)dt +
+ dx dx
(3.52)
r
f (r)
where the function f looks now more complicated than in the AdS Schwarzschild case:
�
� � �d
� �2(d−1)
2 2
2 2
r+
µ
r+
µ
r
r
f (r) = 1 − 1 + 2
+ 2
(3.53)
γ
r+
γ
r+
with
(d − 1)g 2 L2
.
(3.54)
(d − 2)κ2
The horizon is located at r+ . The metric is in this form given in [85]. A more general
discussion of AdS Reissner-Nordström black holes is presented in [8]. With the conventions
used in [85] the t-component of the gauge potential is
�
� �d−2 �
r
At (r) = µ 1 −
.
(3.55)
r+
γ2 =
50
From this one can see that
lim At (r) = µ
(3.56)
r→0
and comparing this with the lines below (3.50) shows that µ is the chemical potential. It
is required that At vanishes at the horizon [98] because of the bifurcation surface of the
∂
Killing vector ∂t
located at the horizon.
Going to Euclidean time the temperature of the AdS Reissner-Nordström black hole
can be obtained in the same manner as for AdS Schwarzschild and one gets
�
�
2 2
(d − 2)r+
µ
1
T =
d−
.
(3.57)
4πr+
γ2
Rescaling the coordinates in the same way as in the AdS Schwarzschild case, one finds that
2
r+ appears in the temperature only in the combination µ2 r+
. This means that the temperature will always depend on the chemical potential and so we have different temperatures
for different chemical potentials. We achieved what we wanted – the chemical potential µ
together with the temperature sets a scale in our theory.
Moreover, it is also possible to go to zero temperature in a continuous way, i.e. to the extremal Reissner-Nordström black hole, which also could not be done for AdS Schwarzschild.
3.6
Linear response
Up to now we have seen how to include finite temperature and a chemical potential on the
field theory side and we have found gravity duals for those. So what happens when we add
a small perturbation to our solutions? As long as the perturbations are small we can work
with linear response theory. Here the basic object one wants to compute is the retarded
Green’s function. It gives the linear dependence of the expectation values of operators on
their sources, i.e. a definition for the retarded Green’s function in frequency space is (cf.
e.g. [85])
(0)
�
�OA �(ω, �k) = GR
(3.58)
OA OB ϕB (ω, k) .
(0)
Here ϕB denotes a bulk field with boundary value ϕB as before.
Our discussion of Green’s functions and transport coefficients in this section is again
based on [85].
Perturbing the boundary values causes a perturbation in the whole bulk and it is
common to make the following separation ansatz for the bulk perturbation
�
ϕA (r) �→ ϕA (r) + ϕ̃A (r)ei(k·�x−ωt) .
(3.59)
Plugging ϕ̃A into the linearized bulk equations of motion yields the equation of motion for
the perturbation. Again we need to impose correct boundary conditions on the perturbation. We will display these conditions for the scalar field case. The condition at the AdS
boundary (r → 0) is the same as in the unperturbed case,
� r �(d−∆)
(0)
ϕ̃A (r) =
ϕ̃A + . . . .
(3.60)
L
3.6 Linear response
51
The boundary condition at the horizon is a bit more complicated. We do not want modes
to come out of the event horizon of the black hole. In order to obtain the retarded Green’s
function instead of the advanced one we require the perturbation to be ingoing at the
horizon. This means that for r → r+ (cf. [85])
 4πω
e−i T log(r−r+ ) (cA + . . . ) , T �= 0 ,
ϕ̃A (r) =
(3.61)
ωL2
ei r−r2+ (c̃ + . . . ) ,
T = 0.
A
The cA , c̃A and the dots denote functions of r that are subleading compared to the exponential function. L2 is the radius of the AdS2 part that arises in the near-horizon geometry
in the extremal case (cf. e.g. [85]). We have seen this near-horizon behavior already in
chapter 2 even for the asymptotically flat case.
Now we can read off the Green’s function
GR
OA OB
=
δ�OA �
(0)
δ ϕ̃B
(1)
2∆A − d δ ϕ̃A
=
.
(0)
L
δ ϕ̃B
(3.62)
The first equality follows from (3.58) and the second from the generalization to d dimensions of (3.39). The first equality is of course not only true for scalar fields but applies to
every kind of perturbation one could think of.
As an example we will show how to compute the electrical conductivity in a 2+1 dimensional field theory. Some real material like Graphene can be described by such a theory at
low energies and therefore it would be interesting to compare the AdS/CFT results with
the ”real world“ measurements in Graphene [85].
The gravitational background we consider is given by the four-dimensional extremal
AdS Reissner-Nordström black hole we discussed before.
According to Ohm’s law the current (here in x-direction) and the electric field are
related by
�Jx � = σEx .
(3.63)
We will only look at the zero momentum case �k = 0. We need to add a perturbation in the
bulk which causes the expectation value of the current in the boundary field theory. At
zero momentum the electric field strength at the boundary is given in terms of the vector
potential via
Ex = −∂t A(0)
(3.64)
x .
So a natural choice for perturbing the background is to add a perturbation Ax (r, t) =
Ãx (r)e−iωt in the Maxwell vector potential.
At this point it is important to mention that at the boundary we have an electric field
but no dynamical photon. This means that in the field theory the U (1) symmetry is a
global symmetry. In the bulk instead the U (1) symmetry is gauged. This is also true in
cases with a different symmetry group than U (1): A global symmetry of the field theory
coresponds to a gauged symmetry in the bulk (cf. e.g. [85]). In chapter 4 we will look at a
52
global SU (2) symmetry on the field theory side. We call this flavor symmetry in analogy
to the global flavor symmetry in QCD.
But let us proceed with the calculation of the conductivity. Equation (3.59) together
with setting the momentum to zero yields
�
�
−iωt
−iωt
Ex = −∂t Ã(0)
e
= iω Ã(0)
= iωA(0)
(3.65)
x
x e
x .
Using Ohm’s law again we can determine the relation between the Green’s function and
the conductivity. As a first step we have
�Jx � = σEx = σ iωA(0)
x .
(3.66)
This is a relation between a source term and an expectation value, so we know from (3.58)
that the proportionality is given by the retarded Green’s function,
(0)
�Jx � = GR
J x J x Ax .
(3.67)
From the last two equations it follows that
i
σ = − GR
.
ω Jx Jx
(3.68)
Now we need to determine the asymptotic behavior of the perturbation close to the boundary at r = 0. To do so one computes the linearized equation of motion for Ax . It can be
shown that close to the boundary the solution looks like (cf. e.g. [85])
Ax (r) = A(0)
x +
r (1)
A + ... .
L x
(3.69)
The constant term to leading order in r yields the source for Jx . We will see shortly how
the expectation value of Jx is related to the next subleading term. From (3.69) we can see
that at the boundary
A(0)
= lim Ax (r) ,
x
r→0
A(1)
x
= lim(LA�x (r)) .
r→0
(3.70)
Here � denotes differentiation with respect to r.
Now we can use (3.37) in order to determine the expectation value of Jx in AdS/CFT:
�
�
δS
f (r) �
1
�Jx � =
= lim
Ax (r) = 2 A(0)
.
(3.71)
(0)
2
r→0
g
g L x
δAx
Here S is the Einstein-Maxwell action (3.49) in four dimensions and the function f was
defined in (3.53). The second equality follows from plugging the background solution into
(0)
the Einstein-Maxwell action and subsequently varying with respect to Ax . More details
can be found in e.g. [85].
3.7 Basic aspects of holographic superconductivity
53
From (3.62) we can compute the Green’s function
GR
J x Jx =
δ�Jx �
(0)
δAx
(1)
=
1 δAx
2
g L δA(0)
x
(3.72)
which yields by virtue of (3.68) the conductivity
(1)
−i δAx
σ(ω) =
.
ωg 2 L δA(0)
x
(3.73)
In general, the conductivity has to be computed numerically, and we will give explicit
examples later. It can be shown that for AdS4 Schwarzschild (i.e. µ = 0) the conductivity
takes the constant value (cf. e.g [99])
σ=
1
.
g2L
(3.74)
The fact that the conductivity in that case is constant was first observed in [100].
Transport coefficients have been calculated with AdS/CFT methods in different setups. One very interesting framework is the construction of possible gravitational duals to
superconductors. We will review this in the next section.
3.7
Basic aspects of holographic superconductivity
”...Aber warum weigern Sie sich nicht, Licht anzudrehen, wenn Sie von Elektrizität nichts
verstehen?”
”...But why do you not refuse to turn on the light if you do not know anything about
electricity?”
”Newton” to the inspector in Friedrich Dürrenmatts ”The Physicists”
Surely the quotation above from Dürrenmatts satiric drama ”The Physicists” has to be
interpreted within the context of the threat of a nuclear war in the 1960ies where a person
who does not know anything about the physical background might be able to fire a nuclear
bomb.
However, it makes one rethink whether we use physical phenomena in technical applications without having a profound knowledge about them. But when can we claim at all
to understand something properly?
It is a similar situation when we consider superconductors. On the one hand it is possible to use superconductors technically, e.g. to generate huge magnetic fields. This is done
for instance at the LHC. On the other hand there are many phenomena in superconductivity which are not understood very well like high temperature superconductors.
54
A powerful theory for describing superconductivity is the BCS theory. In this model the
charge carriers are pairs of electrons (so-called Cooper pairs) bound by the exchange of
phonons, i.e. excitations of the atomic lattice. The spins of electrons in a Cooper pair
have opposite directions which results in a total spin of zero for the Cooper pair (cf. e.g.
[101]). In most superconductors the Cooper pairs have also zero angular momentum and
are therefore called s-wave superconductors.
However, there are superconducting materials which cannot be described by BCS theory. For example, the spins of two electrons forming a Cooper pair can couple into a triplet
state. Then the angular momentum L has to be an odd number due to the requirement
of antisymmetry of the wave function. The possibility with the lowest energy is L = 1.
These kinds of superconductors are then called p-wave superconductors (cf. e.g. [102]).
One approach to gain insight into the mechanism of superconductivity apart from
BSC theory is to construct gravity duals for the field theory describing superconductivity. Within this framework gravitational toy models have been constructed which were
interpreted as the duals of s- and p-wave superconductors.
We will at first present the basic ideas of the holographic description of superconductivity in the case of s-wave superconductors. In chapter 4 we will address in depth p-wave
superconductors. A holographic approach to s-wave superconductors is e.g. presented in
[103, 104, 99]. This section is based on these papers.
For the holographic description of an s-wave superconductor one considers a field theory
in 2+1 dimensions and thence a four-dimensional gravitational dual. This is motivated by
the fact that many unconventional superconductors like the cuprates are layered. Thus an
effective 2+1 dimensional description is adequate (cf. [103]).
The basic idea is to add an additional field to the four-dimensional Lagrangian which
in the case of an s-wave superconductor will be a charged scalar field. The operator dual to
the scalar field develops a non-zero vacuum expectation value below a critical temperature
Tc . This is interpreted as the onset of superconductivity in the field theory. On the gravity
side the scalar field corresponds to a ”hairy” black hole solution.
In four dimensions the no-hair conjecture states that a stationary black hole in asymptotically flat space is completely determined by its mass, charge and angular momentum
(cf. e.g. [3]).
Of course this need not be true for a black hole in asymptotically AdS4 . Here it is well
possible for a black hole to have a ”hair”, which in the case considered in [103, 104] is a
non-trivial profile for the scalar field.
In [104] the following ansatz for the Lagrangian is used:
L=R+
6
1
2
− Fµν F µν + 2 |ψ|2 − |∇ψ − iqAψ|2 .
2
L
4
L
(3.75)
This is the usual Einstein-Maxwell Lagrangian with an additional scalar field which is
complex and carries electric charge. That particular choice can be motivated by LandauGinzburg theory for superconductivity. Here a scalar field is introduced to describe the
Cooper pairs. This field carries electric charge due to the two electrons forming a Cooper
55
pair and therefore it has to be complex (cf. e.g. [92]). In [104] the ansatz
ds2 = −g(r)e−χ(r) dt2 +
dr2
+ r2 (dx2 + dy 2 )
g(r)
(3.76)
for the line element of the hairy black hole is used together with
A = φ(r)dt ,
ψ = ψ(r) .
(3.77)
Note that for this ansatz the radial coordinate ranges over [r+ , ∞[, i.e. the boundary is in
this coordinates located at r = ∞. From the equations of motion it can be derived that
close to the boundary φ and ψ behave as
ρ
+ ... ,
r
ψ (1) ψ (2)
ψ(r) =
+ 2 + ... .
r
r
φ(r) = µ −
(3.78)
There are two different choices of boundary conditions for ψ. Setting one of the ψ (i) to
zero, the other one determines the expectation value of the dual operator Oi (cf. [104]).
There is no source term for the operator Oi and so the symmetry is now spontaneously
broken. In [104] the authors show that above a certain critical temperature Tc the solution
derived from the Lagrangian (3.75) is the Reissner-Nordström black hole with the scalar
field ψ = 0 whereas for lower values of the temperature one of the operators Oi ,i ∈ {1, 2}
develops a non-zero expectation value which means that the scalar field ψ is non-trivial
1
(see also fig. 3.48 ). Furthermore, at T close to Tc one has �Oi � ∼ Tci (1 − TTc ) 2 for i ∈ {1, 2}
which is typical for a second-order phase transition [104].
At the end of the day, one wants to compute the frequency dependent conductivity to see
whether superconductivity appears below Tc . For this purpose one needs to implement the
method described in the previous section. Also for studying the superconducting case one
includes a perturbation Ax with time dependence proportional to e−iωt . The equation of
motion for Ax is again obtained by plugging the perturbation into the linearized equations
of motion. Often one works in a limit in which the scalar and gauge field do not back
react on the metric. This limit is called the probe limit. In [104] the equations are worked
out even with back-reaction but the probe limit was used successfully in the earlier paper
[103]. The asymptotic behavior of the perturbation can be obtained from its equation of
motion and comes out to be (for L = 1)
(1)
Ax (r) = A(0)
x +
8
Ax
+ ...
r
for r → ∞ .
(3.79)
We would like to thank Christopher Herzog for the permission to use the graphics from his homepage.
56
1.2
10
1.0
8
0.8
�O1 �
6
Re�Σ� 0.6
Tc
4
0.4
2
0.2
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0
50
100
T
Ω
Tc
T
150
200
Figure 3.4: Development of the condensate of the operator O1 (on the left) and the
formation of the frequency gap for this operator.
This is the same as (3.69) when taking into account the different radial variables.
The conductivity can now be computed exactly in the same manner as for the nonsuperconducting case. The principle method is the same, but of course the differential
equations one actually has to solve are different ones. Let us recall that
(1)
σ=
−i δAx
.
ωgL2 δA(0)
x
(3.80)
Here g is the coupling constant for the Maxwell field which in (3.75) was absorbed in the
field strength.
Some of the results of [104] and [103] are the following. The imaginary part of the
conductivity has a pole at ω = 0 which leads to a delta function contribution in the
real part of the conductivity due to the Kramers-Kronig relations. This delta function
appears also in the normal state when going away from the probe limit. In the latter
case the delta function is interpreted as a result of translational invariance instead of
superconductivity and the coefficient in front of it differs from the superconducting state.
In the superconducting phase the real part of the conductivity drops for frequencies less
than a gap frequency ωg (see fig. 3.4). Both of these results are consistent with results for
conventional superconductors. A certain amount of energy is needed to break a Cooper pair
and create two independent electrons. Then ωg matches this energy. In [104] the authors
found that the frequency gap matches the BCS result only in the probe limit indicating
that there are different mechanisms at work.
Also the delta function contribution in the real part of the conductivity appears in
conventional superconductors. One way to obtain the conductivity is to calculate it using
the Drude model and then taking the limit of the relaxation time going to infinity. This
procedure also leads to a delta peak in the real part of the conductivity (cf. e.g. [105]).
Having introduced the AdS/CFT correspondence and the holographic approach to field
theoretic problems like the description of s-wave superconductivity, we will see in the next
chapter how these techniques can be applied in a specific setup involving a non-Abelian
57
symmetry group. Also in the later chapters 5 and 6 we will stay within the context of
AdS/CFT. There we will concern ourselves with the attractor mechanism established at
the beginning of chapter 2 and construct some black solutions including such with AdS
asymptotics.
58
Chapter 4
Holographic Flavor Conductivity
In this chapter we investigate some features of the frequency dependent conductivity tensor
in gauge theories dual to four-dimensional Einstein-Yang-Mills theory with SU (2) gauge
group. This theory was used before in [7]in order to describe a p-wave superconductor
holographically.
We find that the Hermitean part of the conductivity tensor in the normal state of the
dual field theory has a negative eigenvalue for a certain range of frequencies. This leads to
a negative entropy production rate.
At the time of writing this thesis we could not give a satisfying explanation of this
result. We will discuss this issue in 4.3.
We have already seen in 3.7 how gravity duals for s-wave superconductors have been constructed. One of the main differences between s-wave and p-wave superconductors is that
in the first case the order parameter that indicates the phase transition to superconductivity is a complex scalar. For a p-wave superconductor instead it is a vector (see [106] for
a summary of p-wave superconductors).
In the construction of a gravitational dual of a p-wave superconductor one is therefore
led to search for black holes with vector hair rather than scalar hair. Such hairy black
holes have been constructed within Einstein-Yang-Mills theory, especially for the case of
an SU (2) gauge group. Results on Yang-Mills black holes in asymptotic AdS space even
with more general gauge groups are reviewed in [4].
In [6, 7] it was shown how a symmetry breaking condensate of a non-Abelian gauge field
arises in four-dimensional Einstein-Yang-Mills theory with gauge group SU (2). In [7] the
SU (2) symmetry is explicitly broken by the background gauge field A = Φ(r)τ 3 dt. Below
a critical temperature the background becomes unstable and the true ground state is given
by the charged black hole with vector hair, for instance in τ 1 direction. This vector hair
breaks the U (1)3 gauge group spontaneously. By looking at fluctuations of the gauge field
around this configuration [7] calculated the conductivity related to the U (1)3 subgroup
and found a gap at low temperature and low frequency as well as a delta function peak
at frequency ω = 0. This was interpreted as a sign of superconductivity in the dual field
theory.
60
4. Holographic Flavor Conductivity
We consider the other components of the conductivity tensor in the normal state of
this theory, in particular those involving currents in the τ 1 and τ 2 directions. Even in the
normal state we find that for small frequencies one of the eigenvalues of the Hermitean
part of the conductivity tensor turns negative.
4.1
Setup
In view of the gauge/gravity correspondence, we consider the four-dimensional EinsteinYang-Mills theory with a cosmological constant
�
�
�
√
1
1 a 2
6
4
S = 2 d x −g R − (Fµν ) + 2 ,
(4.1)
2κ
4
L
where
a
Fµν
= ∂µ Aaν − ∂ν Aaµ + g�abc Abµ Acν ,
µ, ν ∈ {t, r, x, y} ,
(4.2)
is the field strength of an SU (2) gauge group. We are interested in describing a thermal
state of the dual field theory, and, thus, our starting point is a black hole solution of (4.1).
A charge can be introduced into the thermal ensemble by adding a charge for the black
hole. We are only considering the limit of large gL for simplicity. It was argued in [6] that
in this limit the backreaction of the gauge fields on the metric can be neglected. Thus, we
can consider the AdS4 -Schwarzschild black hole metric
ds2 =
�
1 �
1
−f (r)dt2 + r2 (dx2 + dy 2 ) + L2
dr2
2
L
f (r)
(4.3)
with (setting the horizon to rH = 1)
f (r) = r2 −
1
.
r
(4.4)
The temperature of the black hole is then given by (cf. [7])
T =
3
.
4πL2
(4.5)
In addition, the background contains a gauge field
A = Φ(r)τ 3 dt + w(r)τ 1 dx ,
(4.6)
ρLκ2
+ O(1/r2 )
r
(4.7)
where in general (cf. [7])
Φ(r) = µ −
with µ the chemical potential and ρ the charge density of the dual field theory. Moreover,
w(r) in (4.6) describes the condensate that develops when the temperature drops below
4.1 Setup
61
a critical value [7]. The background gauge field in the normal state leaves the U (1)3
subgroup of SU (2) unbroken, which is furthermore broken spontaneously, once the vector
hair w(r) forms. Since [7] was the starting point for our considerations we will review their
calculations here for completeness. So let us for the time being keep the condensate w
although later we will perform our calculations only in the normal state where w = 0.
In [7] the components of the gauge field are rescaled according to
Φ̃ = gL2 Φ ,
w̃ = gL2 w .
(4.8)
It is also convenient to introduce a rescaled chemical potential via
µ̃ = gL2 µ .
(4.9)
The relevant Yang-Mills equations for the rescaled functions are then
2
1
Φ̃�� + Φ̃ −
w̃Φ̃ = 0 ,
3
r
r(r − 1)
1 + 2r3 �
r2
w̃�� +
w̃
+
Φ̃2 w̃ = 0 .
r(r3 − 1)
(r3 − 1)2
(4.10)
The goal is now to solve this coupled system of differential equations together with appropriate boundary conditions for Φ̃ and w̃ both at the horizon and at asymptotic infinity
(i.e. at the boundary of the asymptotically AdS4 spacetime).
It is demanded that at asymptotic infinity the fields Φ̃ and w̃ behave like
Φ̃(r) = p̃0 +
w̃(r) =
p̃1
+ ... ,
r
W̃1
+ ... .
r
(4.11)
Actually, it follows from the Yang-Mills equations that at infinity
w̃(r) = W̃0 +
W̃1
.
r
(4.12)
But since an additional constant term W̃0 in the expansion of w̃ would result in a source
term for the corresponding current such a term is forbidden when one wants to model
spontaneous symmetry breaking. The near-horizon expansion is
Φ̃(r) = Φ̃1 (r − 1) + . . . ,
w̃(r) = w̃0 + w̃2 (r − 1)2 + . . . .
(4.13)
Analogously as for asymptotic infinity, also this form of Φ̃ and w̃ close to the horizon
can be obtained from solving the equations (4.10) in this regime. The coefficient w̃2 is
not independent of w̃0 . To summarize, the method is the following. The general form of
62
the conditions at the boundary and close to the horizon is obtained by solving the YangMills equations in both regimes. These boundary conditions are then supplemented by the
additional requirement of spontaneous symmetry breaking, i.e. that there is no constant
term W̃0 . Then one needs to search for a solution that fulfills both kinds of boundary
conditions.
The coefficients p̃0 and p̃1 are related to the chemical potential and charge density via
(cf. [7])
p̃0 = gL2 µ = µ̃ ,
p̃1 = −gκ2 L3 ρ .
(4.14)
The order parameter is W̃1 since it is related to the expectation value of the SU (2) currents
jia , i ∈ {x, y}1 through (cf. [7])
�jia � ∼ W̃1 δi1 δ1a .
(4.15)
An additional constant term in the expansion of w̃ in (4.11) would correspond to a source
term for the current ji1 . Since there is no source but only a non-zero expectation value for
the current the symmetry is spontaneously broken.
A solution to the described boundary value problem can be obtained using the so-called
shooting technique. With this method one can transform a boundary value problem into
an initial value problem. The latter can easily be solved by Mathematica’s NDSolve. The
first step to start the shooting algorithm is to transform the equations (4.10) into a system
of first order differential equations. The boundary conditions at infinity are then translated
into the problem of finding the zeros of an objective function. We do not want to go into
the technical details here. An instruction how to implement the shooting technique and
some examples can be found in [107]. Basically for the case we consider one includes an
additional constant term W̃0 in the expansion for w̃ (4.11). Then one integrates (4.10)
numerically from the horizon to the boundary using the initial conditions in terms of Φ̃1
and w̃0 . For w̃0 �= 0 this will lead to a non-trivial numerical solution for the condensate
w̃. Of this numerical solution it is required that it obeys (4.11), i.e. that the additional
constant W̃0 vanishes. So what one actually does is to compute W̃0 (Φ̃1 , w̃0 ) and to look
for the zeros of this function. Then one takes the smallest zero (it is argued in [7] that
the larger zeros correspond to thermodynamically disfavoured solutions). The values of Φ̃1
and the corresponding values of w̃0 such that there is no constant term in the expansion
of the condensate are displayed in figure 4.1. This is basically the same as figure 2 in [7].
Figure 4.1 can be interpreted as follows. The normal state with w̃ = 0 is a solution
to the equations of motion (4.10) independently of the chosen value of Φ̃1 . But for values
of Φ̃1 below a certain critical value there exists also a solution to (4.10) with non-zero w̃
which has lower free energy. In [7] this was interpreted as a phase transition from the
normal state to a superconducting one. The critical temperature of the phase transition
can be obtained in the following way.
1
In our notational convention Greek indices run over the bulk indices {t, r, x, y} and latin indices over
field theory indices {t, x, y}.
4.1 Setup
63
5
4
w̃0
3
2
1
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Φ̃1
Figure 4.1: Forming of the symmetry breaking condensate below a critical value for Φ̃1 .
In the normal state the gauge field takes the form
�
�
1
Φ(r) = µ 1 −
,
r
(4.16)
The latter is the form Φ(r) takes in AdS4 Reissner-Nordström (see equation (3.55) which
is also appropriate for considering AdS4 Schwarzschild in the probe limit. From the comparison of (4.7) and (4.16) we read off that in the normal state
µ = ρLκ2 .
(4.17)
Comparing this with (4.14) together with (4.9) leads to the rescaled gauge field in the
normal state
�
�
1
Φ̃(r) = µ̃ 1 −
,
(4.18)
r
i.e. the coefficient Φ̃1 in the near-horizon expansion (4.13) is in the normal state given by
Φ̃1 = µ̃ .
This can be translated into a temperature T̃ in units of
(4.19)
√
µ̃ via
T
T
3
�
T̃ = √ = � =
2
µ̃
Φ̃1
4πL Φ̃1
(4.20)
where in the last step we used (4.5). The second equality is only valid in the normal state.
From figure 4.1 one reads off the value Φ̃1,c ≈ 3.7 at which the phase transition occurs.
3
√
This implies a critical temperature of
. In figure 4.1 in the superconducting state
2
4πL
Φ̃1,c
(i.e. for non-trivial w̃) the rescaled chemical potential µ̃ grows as Φ̃1 is lowered. This
means that in the superconducting state smaller values of Φ̃1 correspond to smaller values
of T̃ .
64
As already mentioned the development of the condensate w̃ was interpreted in [7] as the
phase transition to a superconducting state. This was analyzed in [7] by considering the
conductivity for the current j 3 .
In contrast, we are interested in the conductivities describing the linear response of j 1
and j 2 to electric fields E 1 and E 2 in the normal state of the theory.
In order to obtain the conductivities holographically, we have to look at fluctuations aaµ
of the gauge fields around the background. In general, in spacetimes which are asymptotic
to AdS4 , the gauge fields with zero momentum can be expanded as (cf. (3.79) in the
introduction)
�
�
a,(1)
A
µ
Aaµ = e−iωt Aa,(0)
+
+ O(r−2 ) .
(4.21)
µ
r
a,(0)
Here Aµ is the boundary value of the gauge field. We choose a gauge such that Ar = 0.
We can define the conductivities according to Ohm’s law
jia = σijab Ejb ,
i, j ∈ {x, y} ,
(4.22)
where
a,(0)
Eia = −Fti
b,(0)
= −∂t Aa,(0)
− g�abc At
µ
c,(0)
Ai
.
(4.23)
(0)
There is no term ∂i At in (4.23) since At depends only on r. With this notion of the
conductivity tensor we are interested in σ ab with a, b = 1, 2. In earlier literature the focus
was on σ 33 , which was calculated for the normal state in [108] and for the superconducting
state in [7].
One comment is in order here. In the non-Abelian case, Ohm’s law would rather be
given by (cf. formula (2) of [109])
�
�
jia = σijab Ejb − (Dµ)bj ,
(4.24)
where
d,(0)
(Dµ)bj = ∂j µb − g�bcd µc Aj
.
(4.25)
This is the non-Abelian generalization of the fact that the gradient of the chemical potential
causes a diffusion current. In the Abelian case the current is driven not only by the electric
field but by a field Ẽ = E + 1e ∇µ (cf. e.g. [110]). Of course, in the Abelian case ∇ denotes
the ordinary derivative and not a covariant one.
However, care has to be taken with the expression (4.24). The chemical potential in
formula (2) of [109] is really a dynamical field whereas all quantities coming from the
background, that are imprinted on the field theory from the outside, are included in the
expression Eµb of [109]2 . This is different from the case we consider. What we call the
2
We thank Yaron Oz for discussing this point with us.
4.1 Setup
65
3,(0)
chemical potential, i.e. At
is part of the external field and thus already included in Eia
(cf. (4.23)). No other components of At are sourced by the external sources we are going
to consider.
To determine the conductivities the perturbations aai of the gauge field have to be expanded
according to (4.21). Then the currents caused by linear response to the gauge fields are
a,(0)
a,(1)
given by the following relation between the ai
and ai
in the expansion for aai ,
jia =
2 a,(1)
b,(0)
ai
= −KijR,ab aj ,
2
κ
(4.26)
where
a,(1)
KijR,ab
2 δa
= − 2 ib,(0) .
κ δaj
(4.27)
We used the symbol KijR,ab instead of GR,ab
in order to indicate that (4.27) indeed defines
ij
the response function which in general is different from the retarded Green’s function. This
fact was stressed for instance in [111] and it is well known in the field of condensed matter
physics. The discrepancy between the retarded Green’s function and the response arises
whenever the current itself depends on the gauge field, cf. e.g. the discussion in chapter 7
of [112].
ab
For concreteness, we only consider the components σyy
, a, b = 1, 2, of the conductivity
tensor in the normal state. The relevant coupled equations of motion for a1y and a2y were
already given in sec. 5.1 of [7]. For w̃ = 0 they read
�
�
3
2
2 4
2
2r
+
1
r
(ω
L
+
Φ̃
)
2iωL2 r2 Φ̃ 2
1
∂r2 +
∂
+
a
−
a = 0,
r
y
r(r3 − 1)
(r3 − 1)2
(r3 − 1)2 y
�
�
3
2
2 4
2
2r
+
1
r
(ω
L
+
Φ̃
)
2iωL2 r2 Φ̃ 1
2
∂r2 +
∂
+
a
+
a = 0.
r
y
r(r3 − 1)
(r3 − 1)2
(r3 − 1)2 y
(4.28)
Once the asymptotic form of the solutions is known one can calculate the response functions
according to (4.26). In order to obtain the conductivities, we need to translate this into the
form (4.22). For Abelian gauge fields this is straightforward because of the linear relation
between the gauge potential and the field strength as we have seen in section 3.7.
For a non-Abelian theory things are in general more complicated due to the nonlinear
structure of the field strength (4.23). However, for the case at hand, i.e. with a1y , a2y and A3t
being the only non-vanishing components of the gauge fields, there is still a linear relation
between the field strengths and the fluctuations of the gauge fields, i.e.
�
µ̃
Ey1 = (iωa1y + gΦa2y )�r→∞ = iωa1,(0)
+ 2 a2,(0)
,
y
L y
�
µ̃
Ey2 = (iωa2y − gΦa1y )�r→∞ = iωa2,(0)
− 2 a1,(0)
.
(4.29)
y
L y
66
For ω 2 L4 �= µ̃2 this can be inverted to give
�
�
�
�� 1 �
1,(0)
1
Ey
ay
−iω Lµ̃2
=
.
µ̃
2
2,(0)
µ̃
Ey2
− L2 −iω
ay
ω 2 − L4
(4.30)
Combining this with (4.26), we obtain
� 1 �
� R,11
��
�� 1 �
R,12
1
jy
Kyy
Kyy
Ey
iω − Lµ̃2
=
µ̃
2
2
R,21
R,22
µ̃
jy
Kyy
Kyy
Ey2
iω
ω2 − 4
L2
L
≡
�
11
12
σyy
σyy
21
22
σyy σyy
��
Ey1
Ey2
�
,
(4.31)
where in the last line we defined the conductivity.
It is often more convenient to work with a rescaled conductivity as in [7]. This is
obtained by dividing σ by the value that the conductivity takes for AdS4 -Schwarzschild,
i.e.
σ∞ ≡
2L2
.
κ2
(4.32)
,
(4.33)
Then we define
ab
σ̃yy
=
1 ab
σ
σ∞ yy
a, b = 1, 2 ,
which can be expressed in terms of µ̃ = gL2 µ and
R,ab
K̃yy
=
κ2 R,ab
K
2 yy
(4.34)
as
ab
σ̃yy
1
= 2 4
ω L − µ̃2
�
R,11
R,12
K̃yy
K̃yy
R,21
R,22
K̃yy
K̃yy
��
iωL2 −µ̃
µ̃
iωL2
�
.
(4.35)
Given that the equations of motion (4.28) only depend on the combination ωL2 it is
ab
obvious that σ̃yy
only depends on this dimensionless quantity. Using (4.5) one sees that
2
ωL is proportional to ω/T . In the following section we will compute the conductivities
ab
σ̃yy
as functions of ωL2 . However, in order to avoid cluttering of the formulas, we will set
L = 1. Nevertheless, ω should always be understood as the dimensionless quantity ωL2 .
4.2
Conductivities
ab
In this section we calculate the conductivities σ̃yy
, a, b = 1, 2, both analytically and numerically in the normal state of the field theory.
4.2 Conductivities
4.2.1
67
Analytical solution
In this section we follow the calculation of [113] by expanding the (ingoing) solution for
small µ̃ and ω,
a1y
=
a1(0)
y
�
r−1
r+1
�− iω3 �
2
1,µ̃
1,ω µ̃
1 + ω a1,ω
(r) + ω 2 a1,ω
(r)
y (r) + µ̃ ay (r) + ω µ̃ ay
y
�
2
+ µ̃2 a1,µ̃
(r)
+
.
.
.
(4.36)
y
and similarly for a2y . Plugging this into (4.28) and solving the resulting equations order by
order in the ω and µ̃ expansion (demanding that all functions a1,#
y (r) are regular at the
1(0)
horizon, r = rH = 1, and that they vanish at the boundary, r → ∞, so that a1y → ay
2(0)
and a2y → ay as r → ∞) leads to
a1y
r→∞
−→
a1(0)
y
�
2iω
1+
3r
��
2(0)
iω
2iπ ω µ̃ ay
1+
− √
+
3r 3 3 r a1(0)
y
2(0)
a2y
√
3π − 3 ln 3 µ̃2
6
r
�
ω 2 µ̃ ay
ω 2 µ̃2
+c1
+
c
+ ... ,
2
r a1(0)
r
y
√
�
��
1(0)
2iω
iω
2iπ ω µ̃ ay
3π − 3 ln 3 µ̃2
r→∞
2(0)
−→ ay
1+
1+
+ √
+
3r
3r 3 3 r a2(0)
6
r
y
�
1(0)
ω 2 µ̃ ay
ω 2 µ̃2
−c1
+
c
+ ... ,
2
r a2(0)
r
y
(4.37)
with c1 and c2 two real constants that we could not determine analytically, but could read
off from the numerical results of the next section. We left out a term of the order O(ω µ̃2 ),
as it is always subleading with respect to the term proportional to ω (note that it would
also come with a factor of i, being linear in ω). Moreover, there is no ω 2 -term (or higher
order in ω without any µ̃-factor). This can be understood because in the limit µ̃ = 0 the
equations can be solved explicitly and the expansion of the solution only has a term linear
in ω at order 1/r, cf. formula (38) of [7].
From (4.26) and (4.37) we read off
√
3π − 3 ln 3 2
R,11
R,22
K̃yy (k = 0, ω) = −iω −
µ̃ + c2 ω 2 µ̃2 . . . = K̃yy
(k = 0, ω) ,
6
2iπ
R,12
R,21
K̃yy
(k = 0, ω) = √ ω µ̃ + c1 ω 2 µ̃ + . . . = −K̃yy
(k = 0, ω) .
(4.38)
3 3
The relations
R,22
R,11
K̃yy
= K̃yy
,
R,21
R,12
K̃yy
= −K̃yy
,
(normal phase) ,
(4.39)
68
are consistent with formula (4.60) of [113]. Plugging (4.38) into (4.31), we obtain to leading
order in ω and µ
ω2
0.8512µ̃2 ω
22
+
i
+ . . . = σ̃yy
,
ω 2 − µ̃2
ω 2 − µ̃2
0.3576µ̃3 − 1.2092µ̃ω 2
µ̃ω
21
=
+i 2
+ . . . = −σ̃yy
,
ω 2 − µ̃2
ω − µ̃2
11
σ̃yy
=
(4.40)
12
σ̃yy
(4.41)
where we also used
√
4.2.2
3π − 3 ln 3
≈ 0.3576 .
6
(4.42)
Numerics
Numerically, we do not have to restrict to small values of ω and µ̃. In order to be able
to compare with the analytical results of the last section, let us nevertheless start by
solving (4.28) numerically for small values of ω and for µ̃ = 0.01. Let us first outline how
to calculate the retarded Green’s functions numerically. We follow closely [7]. The near
horizon expansion is
a1y = (r − 1)−iω/4πT [β1 + α1 (r − 1) + . . .] ,
a2y = (r − 1)−iω/4πT [β2 + α2 (r − 1) + . . .] ,
(4.43)
where β1 and β2 are two independent constants which correspond to the two free constants
of integration for (4.28) which are left after specifying the solution to be ingoing at the
horizon.3 The other coefficients in the expansion (4.43) are determined by β1 and β2 . For
α1 and α2 we find
α1 = −
2β2 ω Φ̃1
9i + 6ω
,
α2 =
2β1 ω Φ̃1
,
9i + 6ω
(4.44)
where Φ̃1 is the coefficient in the near horizon expansion. Let us recall for convenience that
close to the horizon the expansion of Φ̃ is given by (cf. (4.13))
Φ̃ = Φ̃1 (r − 1) + . . . .
(4.45)
Now notice the following. The equations (4.28) consist of two different types of terms.
The ones which do not mix a1y with a2y are even under a change Φ̃ �→ −Φ̃ and the terms
mixing a1y with a2y are odd under Φ̃ �→ −Φ̃. As the equations (4.28) are linear, we can
3
In contrast to the situation in sec. 4.2 of [7], there are no constraints from the linearized Yang-Mills
equations and there is no residual gauge invariance, which reduced the number of independent solutions
in that case.
4.2 Conductivities
69
R,11
Im(K̃yy
)
R,11
Re(K̃yy
)
ω
1
0.00001
2
3
4
!1
1
2
3
ω
!0.00001
4
!2
!3
!0.00002
!4
!0.00003
R,12
Re(K̃yy
)
R,12
Im(K̃yy
)
ω
1
2
0.006
3
4
0.005
!0.002
0.004
!0.004
0.003
!0.006
0.002
!0.008
0.001
ω
!0.010
1
2
3
4
R,11
R,11
R,12
R,12
Figure 4.2: Re(K̃yy
), Im(K̃yy
), Re(K̃yy
) and Im(K̃yy
) as a function of ω for µ̃ = 0.01.
expand the asymptotic solutions as
1,(1)
ay
β1 a1,(1),e + β2 a1,(1),o
+ . . . = β1 a1,(0),e + β2 a1,(0),o +
+ ... ,
r
r
2,(1)
ay
β2 a2,(1),e + β1 a2,(1),o
2,(0),e
2,(0),o
= a2,(0)
+
+
.
.
.
=
β
a
+
β
a
+
+ ... ,
2
1
y
r
r
a1y = a1,(0)
+
y
a2y
(4.46)
where a1,(0),e , a1,(0),o , a1,(1),e , a1,(1),o , a2,(0),e , a2,(0),o , a2,(1),e , a2,(1),o are constants (depending on
ω and µ̃) which are either even (e) or odd (o) under Φ̃ �→ −Φ̃, indicated by their superscript.
In order to calculate the response functions, we make use of formula (4.26). We notice
that
�
1,(0)
ay
2,(0)
ay
�
=
�
a1,(0),e a1,(0),o
a2,(0),o a2,(0),e
��
β1
β2
�
,
(4.47)
which can be inverted to give
�
β1
β2
�
=
�
a1,(0),e a1,(0),o
a2,(0),o a2,(0),e
�−1 �
1,(0)
ay
2,(0)
ay
�
.
(4.48)
70
R,11
Re(K̃yy
)
R,11
Im(K̃yy
)
0.02
!0.00003575
ω
0.04
0.06
0.08
0.10
!0.02
!0.0000358
!0.00003585
!0.04
!0.0000359
!0.06
!0.00003595
!0.08
0.02
0.04
R,12
Re(K̃yy
)
0.02
0.06
ω
0.08
0.06
!0.10
R,12
Im(K̃yy
)
ω
0.04
0.10
0.0012
0.08
0.10
0.0010
!0.00002
0.0008
!0.00004
0.0006
!0.00006
0.0004
!0.00008
0.0002
ω
!0.0001
0.02
0.04
0.06
0.08
0.10
R,11
R,11
R,12
R,12
), Im(K̃yy
), Re(K̃yy
) and Im(K̃yy
This then leads to
�
�
� 1,(1),e
1,(1)
ay
a
=
2,(1)
a2,(1),o
ay
� 1,(1),e
a
=
a2,(1),o
a1,(1),o
a2,(1),e
a1,(1),o
a2,(1),e
��
��
β1
β2
�
a1,(0),e a1,(0),o
a2,(0),o a2,(0),e
(4.49)
�−1 �
1,(0)
ay
2,(0)
ay
�
from which one can read off the retarded response functions
� R,11
�
� 1,(1),e 1,(1),o � � 1,(0),e 1,(0),o �−1
R,12
2
Kyy
Kyy
a
a
a
a
=− 2
.
R,21
R,22
2,(1),o
2,(1),e
2,(0),o
Kyy
Kyy
a
a
a
a2,(0),e
κ
(4.50)
(4.51)
Notice that the coefficients a1,(0),e etc. are independent of β1 and β2 and, thus, can be
calculated for arbitrary (non-vanishing) choices. The coefficients a1,(0),e etc. can easily be
calculated numerically, by solving (4.28) for a background Φ̃ and for −Φ̃ and then taking
sums or differences of the results in an obvious way.
R,ab
The result for K̃yy
is displayed in fig. 4.2 and a zoom into the region of small ω
values is given in fig. 4.3 (this will be important to compare with the analytic result of
R,11
R,12
the previous section). We only show K̃yy
and K̃yy
, as these are the only independent
quantites in the normal phase, due to (4.39).
The result agrees nicely with (4.38) for c1 ≈ −1.15 and c2 ≈ −0.28 and using (4.42).
R,11
The value of K̃yy
in the limit of ω = 0 depends on µ̃ and one can easily obtain this
dependence for larger values of µ̃ numerically, cf. fig. 4.4. Obviously it diverges at the phase
transition to the superconducting phase which occurs at µ̃ = Φ1,c . This is understandable
4.2 Conductivities
71
R,11
Re(K̃yy
)(ω = 0)
0.5
1.0
µ̃
1.5
2.0
2.5
3.0
3.5
!5
!10
!15
R,11
)(k = 0, ω = 0) as a function of µ̃.
from the analysis of the quasinormal mode spectrum performed in sec. 5.1 of [7]. At the
phase transition a Goldstone mode appears in the spectrum of modes involving a1y . This
R,11
leads to a pole at ω = 0 in K̃yy
.
Using (4.31), we can calculate the conductivity and find the result displayed in fig. 4.5.
Again we note that
22
11
σ̃yy
= σ̃yy
,
21
12
σ̃yy
= −σ̃yy
.
(4.52)
It is even more instructive to look at the eigenvalues of the Hermitean part
1 ab
ba ∗
Σab
h = (σ̃yy + (σ̃yy ) )
2
(4.53)
and the anti-Hermitean part
Σab
ah =
1 ab
ba ∗
(σ̃ − (σ̃yy
)).
2i yy
(4.54)
They are plotted in fig. 4.6.
Some comments are in order. Obviously the matrix Σab is not positive semi-definite.
Its eigenvalues go to zero, though, in the DC limit ω → 0. Moreover, the conductivity
shows a characteristic pole at ω = µ̃ which arises due to the denominator in equation
(4.31). Finally note that all conductivities go to their values in AdS-Schwarzschild (without
chemical potential) for large values of ω, as expected.
Let us make a few more comments here. The fact that the matrix Σ develops a negative
eigenvalue for ω < µ̃ indicates that the entropy production rate becomes negative for that
frequency range, potentially leading to an instability. This can be checked explicitly. The
entropy production rate is generally given by
�
�
�
� a )∗ = −Re (K̃ R,11 a1,(0) + K̃ R,12 a2,(0) )(iωa1,(0) + µ̃a2,(0) )∗
ṡ = Re �j a · (E
yy
y
yy
y
y
y
�
R,21 1,(0)
R,22 2,(0)
+(K̃yy
ay + K̃yy
ay )(iωa2,(0)
− µ̃a1,(0)
)∗ ,
(4.55)
y
y
72
11
Re(σ̃yy
)
11
Im(σ̃yy
)
4
0.02
3
0.01
2
1
0
0.00
0.02
0.04
0.06
0.08
0.10
ω
!1
!2
0.02
0.04
0.06
0.08
0.10
ω
!0.01
!0.02
12
Re(σ̃yy
)
12
Im(σ̃yy
)
2.0
0.02
1.5
0.01
ω
0.00
0.02
0.04
0.06
1.0
0.08
0.10
0.5
0.0
!0.01
0.02
0.04
!0.5
!0.02
0.06
ω
0.08
0.10
!1.0
11
11
12
12
Figure 4.5: Re(σ̃yy
), Im(σ̃yy
), Re(σ̃yy
) and Im(σ̃yy
where s is the entropy density. It turns out that the eigenvector of Σ corresponding to
the negative eigenvalue is given by (i, 1), independently of ω. In other words, there is a
relation between the two electrical fields, Ey1 = iEy2 . Comparing this with (4.29), we see
that this can be achieved if
a1,(0)
= ia2,(0)
,
y
y
(4.56)
i.e. the two gauge fields differ by π/2 in phase. From (4.28) and (4.43) we see that choosing
the initial conditions β1 = iβ2 at the horizon will lead, for the normal state of the dual
field theory, to a solution which fulfills a1y = ia2y throughout (and, thus, to a solution with
(4.56)). In that case both equations of (4.28) reduce to the same equation for, say, a1y .
Now, using (4.56) in (4.55), we obtain
�
�
2
R,12
R,11
ṡ = 2(ω − µ̃)|a2,(0)
|
Re
K̃
−
Im
K̃
.
(4.57)
y
yy
yy
The factor (ω − µ̃) comes from the fact that for the gauge fields (4.56) the field strengths
are given by
Ey1 = iEy2 = −a2,(0)
(ω − µ̃) .
y
(4.58)
Obviously, they vanish at ω = µ̃, where σ̃ has its pole. Thus, the entropy production rate
stays finite there (it actually vanishes). Fig. 4.7 shows the entropy production rate for
β1 = iβ2 = i and for µ̃ = 0.01. The change of sign at ω = µ̃ arises because the electrical
field strengths change their sign, whereas the term in brackets in (4.57) does not. Rather
4.3 Discussion of our results
73
EV1 (Σh )
EV2 (Σh )
0.8
4
0.6
2
0.4
0
0.2
!2
ω
0.02
0.04
!0.002
0.02
0.08
0.06
0.08
0.10
ω
0.06
0.08
0.10
EV2 (Σah )
0.04
ω
0.10
0.04
!0.004
0.02
!0.006
0.00
!0.008
0.04
!4
0.06
EV1 (Σah )
0.000
0.02
!0.02
0.02
0.04
0.06
0.08
0.10
ω
!0.010
!0.04
!0.012
ab
Figure 4.6: Eigenvalues of the matrices Σab
h and Σah , given in (4.53) and (4.54), as a
function of ω for µ̃ = 0.01.
it is positive, going to zero for ω → 0. Obviously, the problem with the negative entropy
production rate arises because the currents jy1 and jy2 stay finite even though the field
strengths Ey1 and Ey2 vanish and change sign at ω = µ̃. The negative entropy production
rate might indicate an instability of the background, sourced by the electrical field. We
will discuss this in more detail in the next section.
4.3
Discussion of our results
First of all it is important to notice that the theory seems well behaved from the field
R,ab
theory point of view. This becomes clear when we look at the response function K̃yy
instead of the conductivity. From the discussion in section 2 of [7] it follows that the antiHermitean part of the response function has to be negative semi-definite. In figure 4.8 the
R,ab
eigenvalues of the anti-Hermitean part of K̃yy
are plotted for the range in ω in which the
negative eigenvalue of the Hermitean part of the conductivity appears. The eigenvalues of
R,ab
the anti-Hermitean part of the response function are indeed negative implying that K̃yy
is negative semi-definite.
One possibility to see whether there is an instability on the gravity side is to take into
account backreaction. The full backreacted equations of motion to linear order can be
found in [114]. Their equations (3.12a) to (3.12c) show that the perturbations a1y and a2y
we consider do not source metric perturbations in the normal state and therefore it is not
very likely that including backreaction at the linearized level could cure the problem.
At the moment the most promising idea is to look at the full Einstein-Yang-Mills
74
ṡ
0.0004
0.0003
0.0002
0.0001
ω
0.005
0.010
0.015
0.020
Figure 4.7: Entropy production rate ṡ, eq.(4.57), as a function of ω for µ̃ = 0.01.
ω
0.02
0.04
0.06
ω
0.08
0.10
0.02
�0.02
�0.02
�0.04
�0.04
�0.06
�0.06
�0.08
�0.08
�0.10
�0.10
EV1
0.04
0.06
0.08
0.10
EV2
R,ab
Figure 4.8: Eigenvalues of the anti-Hermitean part of the response function K̃yy
as a
function of ω for µ̃ = 0.01.
equation instead
ones. The reason for that is the following.
� of the linearized
�
a
a
∗
�
For ṡ = Re �j · (E ) we have according to (4.29) and (4.26)
µ̃ 2,(0)
a
,
L2 y
µ̃
= iωa2,(0)
− 2 a1,(0)
y
L y
Ey1 = iωa1,(0)
+
y
Ey2
as well as
a,(1)
jia ∼ ai
.
(4.59)
(4.60)
But this means that the entropy production rate ṡ depends quadratically on the perturbations a1y and a2y . Such an effect cannot be seen directly at the linearized level in perturbation
theory. We therefore hope that by considering the full Einstein-Yang-Mills equations we
might see directly that the horizon area decreases in the presence of non-Abelian sources.
This would clearly indicate an instability on the gravity side.
Chapter 5
Nernst branes in four-dimensional
gauged supergravity
In this chapter we study static black brane solutions in the context of N = 2 U (1) gauged
supergravity in four dimensions with only vector multiplets. Using the formalism of firstorder flow equations, we construct novel extremal black brane solutions including examples
of Nernst branes, i.e. extremal black brane solutions with vanishing entropy density. This
chapter is based on [1].
In this chapter we are particularly interested is black solutions which asymptote to AdS4 .
As we know from our discussion of the AdS/CFT correspondence in chapter 3 black solutions in an asymptotically AdS space represent gravity duals for field theories at finite
temperature. Therefore exploring the rich variety of black geometries might lead to various
field theoretic applications.
We are especially interested in extremal black brane solutions. Firstly, as we have
mentioned in chapter 2, in flat space it is impossible to have a near-horizon geometry that
is not a sphere. Therefore black branes, i.e. objects with flat near-horizon geometry like
R2 naturally arise in asymptotically non-flat backgrounds such as AdS4 . Secondly, the
great thing about extremal black objects is that the attractor mechanism is at work as we
have explained in chapter 2. Furthermore, extremality comes accompanied with vanishing
temperature (cf. chapter 2). This is important when one wants to construct solutions that
obey the Nernst law.
We will construct extremal black branes with vanishing entropy density. As those fulfill
the Nernst law known from thermodynamics we call them Nernst branes. Prior, smooth
Nernst configurations in AdS have been found in [115, 116, 117].
As already mentioned extremal black geometries exhibit the attractor mechanism. In
chapter 2 we presented the attractor mechanism for the ungauged case. In the next two
chapters we consider the more complicated case of gauged supergravity instead. This
means that we will also include fluxes in the theory (cf. chapter 2). At first, we will
focus on the four-dimensional case. In the presence of fluxes, extremal supersymmetric
configurations in four dimensions were first discovered by [118] and subsequently discussed
76
5. Nernst branes in four-dimensional gauged supergravity
in [119, 120]. The first-order formalism in the presence of fluxes was first presented in
[119]. More results on generalizing the attractor mechanism to gauged supergravity are
presented in [121, 122, 118, 123, 119, 120, 124].
In this setup of N = 2 U (1) gauged supergravity we find new black solutions. Among
them are not only the aforementioned Nernst branes but also non-Nernst configurations.
We find e.g. a solution that generalizes a solution presented in [9] as well as a numerical
solution that interpolates between AdS2 × R2 and AdS4 .
But let us begin with the derivation of the first-oder flow equations.
5.1
Flow equations for extremal black branes in four
dimensions
First-order flow equations for supersymmetric black holes and black branes were recently
obtained in [119] by a rewriting of the action, where they were given in terms of physical
scalar fields z i = Y i /Y 0 (i = 1, . . . , n). Here, we will re-derive them by working in big
moduli space, so that the resulting first-order flow equations will now be expressed in
terms of the Y I . The formulation in big moduli space becomes particularly useful when
discussing the coupling to higher-derivative curvature terms [125].
5.1.1
First-order flow equations in big moduli space
Following [118, 119, 120] we make the ansatz for the black brane line element,
�
�
ds2 = −e2U dt2 + e−2U dr2 + e2ψ (dx2 + dy 2 ) ,
(5.1)
where U = U (r) , ψ = ψ(r). The black brane will be supported by scalar fields that only
depend on r.
The Lagrangian we will consider is given in (A.25). It is written in terms of fields X I
of big moduli space. The rewriting of this Lagrangian as a sum of squares of first-order
flow equations will, however, not be in terms of the X I , but rather in terms of rescaled
variables Y I defined by
Y I = eA X̃ I = eA ϕ̄ X I .
(5.2)
Here A = A(r) denotes a real factor that will be determined to be given by
A=ψ−U ,
(5.3)
while ϕ̄ denotes a phase with a U (1)-weight that is opposite to the one of X I . Thus,
the X̃ I = ϕ̄X I denote homogeneous coordinates that are U (1) invariant and satisfy (A.3)
(with X I replaced by X̃ I ) as well as
¯ �J ,
NIJ Dr X I Dr X̄ J = NIJ X̃ �I X̃
(5.4)
77
where X̃ �I = ∂r X̃ I . Observe that in view of (A.3),
e2A = −NIJ Y I Ȳ J ,
and that
�
�
e2A A� = − 12 NIJ Y �I Ȳ J + Y I Ȳ �J ,
(5.5)
(5.6)
where we used the second homogeneity equation of (A.1).
We will first discuss electrically charged extremal black branes in the presence of electric
fluxes hI only, so that for the time being the flux potential (A.30) reads
�
�
¯ = N IJ − 2 X̃ I X̃
¯J h h .
V (X̃, X̃)
I J
(5.7)
Subsequently, we will extend the first-order rewriting to the case of dyonic charges as well
as dyonic fluxes.
We take FtrI = E I (r) as well as X I = X I (r). Inserting the line element (5.1) into the
action (A.25) yields the one-dimensional Lagrangian
L1d =
√
−g L − QI E I ,
(5.8)
where L is given in (A.25) and the QI denote the electric charges. Extremizing with respect
to E I yields
− e2ψ−2U ImNIJ E J = QI ,
(5.9)
and hence
�
�IJ
E I = −e2U −2ψ (ImN )−1
QJ .
(5.10)
The associated one-dimensional action reads,
�
�
¯ �J − 1 e2U −4ψ Q �(ImN )−1 �IJ Q
− S1d =
dr e2ψ U �2 − ψ �2 + NIJ X̃ �I X̃
I
J
2
�
¯
+g 2 e−2U V (X̃, X̃)
�
�
d � 2ψ
+ dr
e (2ψ � − U � ) ,
(5.11)
dr
in accordance with [119] for the case of black branes. Next, we rewrite (5.11) in terms of
the rescaled variables Y I . We also find it convenient to introduce the combination
�
�
qI = eU −2ψ+iγ QI − i g e2(ψ−U ) hI ,
(5.12)
where γ denotes a phase which can depend on r. Using
�
�
X̃ �I = e−A Y �I − A� Y I ,
(5.13)
78
as well as (5.6), we obtain the intermediate result
�
�
− S1d =
dr e2ψ U �2 − ψ �2
�
��2
�
��
� �
+e−2A NIJ Y �I − eA N IK q̄K Ȳ �J − eA N JL qL + A� + Re X̃ I qI
� �
��2
�
�
�IJ
¯
− 12 e2U −4ψ QI (ImN )−1
QJ − qI N IJ q̄J − Re X̃ I qI
+ g 2 e−2U V (X̃, X̃)
�
�
� �
�
d � 2ψ
2ψ
�I
+2 dr e Re X̃ qI + dr
e (2ψ � − U � ) .
(5.14)
dr
Next, using the identity,
�
�
¯ J + X̃ J X̃
¯I ,
− 1 (ImN )−1 IJ = N IJ + X̃ I X̃
(5.15)
2
as well as the explicit form of the potential (5.7), we obtain
�
�
− S1d =
dr e2ψ U �2 − ψ �2
�
��2
�
��
� �
+e−2A NIJ Y �I − eA N IK q̄K Ȳ �J − eA N JL qL + A� + Re X̃ I qI
� �
��
�
¯ J − Re X̃ I q 2 − 2 g 2 e−2U h X̃ I h X̃
¯J
+2 e2U −4ψ QI X̃ I QJ X̃
I
I
J
�
�
� �
�
d � 2ψ
+2 dr e2ψ Re X̃ �I qI + dr
e (2ψ � − U � ) .
(5.16)
dr
Inserting the expression (5.12) into the fourth line of (5.16) yields
�
�
�
�
�
�
�
��
d � U
2ψ
�I
2 dr e Re X̃ qI = 2 dr
e Re eiγ X̃ I QI + g e2ψ−U Im eiγ X̃ I hI
dr
�
�
�
−2
dr eU U � Re eiγ X̃ I QI
�
�
�
−2 g
dr e2ψ−U (2ψ � − U � ) Im eiγ X̃ I hI
(5.17)
�
�
�
�
�
��
+2
dr γ � eU Im eiγ X̃ I QI − g e2ψ−U Re eiγ X̃ I hI
.
Combining the terms proportional to ψ �2 and to ψ � into a perfect square, and the terms
proportional to U �2 and to U � into a perfect square, leads to
where
SBPS =
− S1d = SBPS + STD ,
�
��
(5.18)
�
�
�
� �2
dr e
U � − eU −2ψ Re eiγ X̃ I QI + g e−U Im eiγ X̃ I hI
�
�
��2
− ψ � + 2 g e−U Im eiγ X̃ I hI
(5.19)
�
�
��2
� �I
� � �J
� � �
−2A
A
IK
A
JL
I
+e
NIJ Y − e N q̄K Ȳ − e N qL + A + Re X̃ qI
+∆ ,
2ψ
and
Finally,
STD =
�
�
�
�
��
∆ = 2 eU −2ψ Im eiγ X̃ I QI − g e−U Re eiγ X̃ I hI
�
�
�
�
��
γ � + eU −2ψ Im eiγ X̃ I QI + g e−U Re eiγ X̃ I hI
.
�
dr
79
(5.20)
�
�
�
��
d � 2ψ
e (2ψ � − U � ) + 2 eU Re eiγ X̃ I QI + 2 g e2ψ−U Im eiγ X̃ I hI
.
dr
(5.21)
Setting the squares in SBPS to zero gives
�
�
�
�
U � = eU −2ψ Re eiγ X̃ I QI − g e−U Im eiγ X̃ I hI ,
�
�
ψ � = −2 g e−U Im eiγ X̃ I hI ,
�
�
A� = − Re X̃ I qI ,
Y �I = eA N IK q̄K ,
while demanding the variation of ∆ to be zero yields
�
�
�
�
eU −2ψ Im eiγ X̃ I QI − g e−U Re eiγ X̃ I hI = 0
as well as
�
γ = −e
U −2ψ
�
iγ
I
�
Im e X̃ QI − g e
−U
�
iγ
I
Re e X̃ hI
�
(5.22)
(5.23)
.
(5.24)
Note that the first-order flow equations for the Y I and for A are consistent with one
another: the latter is a consequence of the former by virtue of (5.6). The flow equations
given above are obtained by varying (5.19) with respect to the various fields and setting
the individual terms to zero. Therefore, the solutions to these flow equations describe a
subclass, but certainly not the entire class of solutions to the equations of motion stemming
from the one-dimensional action (5.11).
Comparing the flow equations (5.22) with the ones obtained in the supersymmetric
context in [119] shows that the flow equations derived above are the ones for supersymmetric black branes, and that the phase γ is to be identified with the phase α of [119] via
γ � = − (α� + Ar ), with Ar given in (A.20).
Next, we study the dyonic case, with charges (QI , P I ) and fluxes (hI , hI ) turned on.
The above results can be easily extended by first writing the term Q(ImN )−1 Q in the
action (5.11) as
�
�IJ
VBH = − 21 QI (ImN )−1
QJ
�
�
¯J Q Q
= N IJ + 2 X̃ I X̃
I
= g ij̄ Di Z D̄j̄ Z̄ + |Z|2 ,
J
(5.25)
80
where we used (A.29) to write VBH in terms of Z = −QI X I . Turning on magnetic charges
amounts to extending Z to [38]
�
�
Z = P I FI − QI X I = P I FIJ − QJ X J = −Q̂I X I ,
(5.26)
where
Q̂I = QI − FIJ P J .
(5.27)
Similarly, the flux potential with dyonic fluxes can be obtained from the one with purely
electric fluxes by the replacement of hI by
ĥI = hI − FIJ hJ ,
(5.28)
cf. (A.30).
Thus, formally the action looks identical to before, and we can adapt the computation
given above to the case of dyonic charges and fluxes by replacing QI and hI with Q̂I and
ĥI . Performing these replacements in (5.12) as well yields
�
�
qI = eU −2ψ+iγ Q̂I − i g e2(ψ−U ) ĥI .
(5.29)
The above procedure results in
− S1d = SBPS + STD + Ssympl ,
(5.30)
where SBPS and STD are given as in (5.19) and (5.21), respectively, with QI and hI replaced
by Q̂I and ĥI , and with qI now given by (5.29). The third contribution, Ssympl , is given by
�
�
�
Ssympl = g
dr QI hI − P I hI .
(5.31)
Observe that this term is constant, independent of the fields, and hence it does not contribute to the variation of the fields. Imposing the constraint S1d = 0 (which is the
Hamiltonian constraint, to be discussed below) on a solution yields the condition
QI h I − P I hI = 0 ,
(5.32)
in agreement with [119] for the case of black branes. The condition (5.32) can also be
written as
�
�
¯
Im Q̂I N IJ ĥJ = 0 .
(5.33)
The flow equations are now given by
�
�
�
�
U � = eU −2ψ Re eiγ X̃ I Q̂I − g e−U Im eiγ X̃ I ĥI ,
�
�
ψ � = −2 g e−U Im eiγ X̃ I ĥI ,
�
�
A� = − Re X̃ I qI ,
Y �I = eA N IJ q̄J ,
�
�
�
�
γ � = eU −2ψ Im eiγ Z̃ + g e−U Re eiγ W̃ ,
(5.34)
81
where Z̃ and W̃ denote Z and W with X replaced by X̃, as in (A.31). Inspection of the
flow equations (5.34) yields
A� = (ψ − U )� ,
(5.35)
and hence we obtain (5.3) (without loss of generality).
Observe that, as before, the flow equations for A and Y I are consistent with one another.
The latter can be recast into
� I
�
� iγ IK
�
� iγ IK
�
(Y − Ȳ I )�
e N Q̂K
e N ĥK
−ψ
ψ−2U
= −2i e Im iγ
+ 2i g e
Re iγ
(,5.36)
(FI − F̄I )�
e F̄IK N KJ Q̂J
e F̄IK N KJ ĥJ
where here FI = ∂F (Y )/∂Y I . Each of the vectors appearing in this expression transforms
as a symplectic vector under Sp(2(n + 1))transformations, i.e. as
� I�
� I
� � J�
Y
A J B IJ
Y
�→
,
(5.37)
FI
CIJ DI J
FJ
where AT D − C T B =
n+1 .
For instance, under symplectic transformations,
¯
¯
N IJ ĥJ �→ S I K N KL ĥL ,
�
�
FIJ �→ DI L FLK + CIK [S −1 ]K J ,
(5.38)
where S I J = AI J + B IK FKJ [126]. Using this, it can be easily checked that the last vector
in (5.36) transforms as in (5.37).
The constraint (5.23) becomes
�
�
�
�
eU −2ψ Im eiγ Z(Y ) − g e−U Re eiγ W (Y ) = 0 ,
(5.39)
where
Z(Y ) = P I FI (Y ) − QI Y I ,
W (Y ) = hI FI (Y ) − hI Y I .
(5.40)
Using the flow equations (5.34), we find that the constraint (5.39) is equivalent to the
condition
qI Y I = q̄I Ȳ I .
(5.41)
The phase γ is not an independent degree of freedom. It can be expressed in terms of
Z(Y ) and W (Y ) as [119]
e−2iγ =
Z(Y ) − ig e2(ψ−U ) W (Y )
.
Z̄(Ȳ ) + ig e2(ψ−U ) W̄ (Ȳ )
(5.42)
When γ = kπ (k ∈ Z), equation (5.42) implies
Z(Y ) = Z̄(Ȳ ) ,
W (Y ) = −W̄ (Ȳ ) .
(5.43)
82
Differentiating (5.39) with respect to r yields a flow equation for γ � that has to be
consistent with the last flow equation in (5.34). We proceed to check consistency of these
two flow equations. Multiplying (5.39) with exp (2ψ − U ) and differentiating the resulting
expression with respect to r yields
�
�
��
Im i qI Y I γ � − 2g e−ψ eiγ W (Y ) = 0 ,
(5.44)
where we used the flow equations for Y I and for (ψ − U ). Using (5.41), this results in
�
�
γ � = 2g e−ψ Re eiγ W (Y ) ,
(5.45)
which, upon using (5.39), equals the flow equation for γ given in (5.34). Thus, we conclude that upon imposing the constraint (5.39), the flow equation for γ � given in (5.34) is
automatically satisfied.
Summarizing, we obtain the following independent flow equations. Since e2A is expressed in terms of Y I , it is not an independent quantity, cf. (5.5). As U = ψ − A on a
solution, U is also not an independent quantity. The independent flow equations are thus
�
�
ψ � = 2 g e−ψ Im eiγ W (Y ) ,
Y �I = eψ−U N IK q̄K ,
(5.46)
with qK given in (5.29) and with γ given in (5.42) (observe that the latter is, in general,
r-dependent). In addition, a solution to these flow equations has to satisfy the reality
condition (5.41) as well as the symplectic constraint (5.32). The flow equations for the
scalar fields Y I can equivalently be written in the form (5.36). Observe that in view of
(5.5) and (5.41), the number of independent variables in the set (U, ψ, Y I ) is the same as
in the set (U, ψ, z i = Y i /Y 0 ), which was used in [119]. Indeed, the flow equations (5.46)
are equivalent to the ones presented there and, thus, they describe supersymmetric brane
solutions.
Observe that the right hand side of the flow equations (5.46) may be expressed in terms
of the Y I only by redefining the radial variable into ∂/∂τ = eψ ∂/∂r, in which case they
become
�
�
∂ψ
= 2 g Im eiγ W (Y ) ,
∂τ
�
�
∂Y I
¯
¯
= e−iγ N IK Q̂K + i g e2A ĥK ,
∂τ
(5.47)
with A expressed in terms of the Y I according to (5.5).
Let us now discuss the Hamiltonian constraint mentioned above. For a Lagrangian
√
density −g ( 12 R + LM ) it is given by the variation of the action w.r.t. g 00 as
1
R
2 00
+
�
�
δLM
1
1
−
g
R
+
L
= 0.
M
2 00 2
δg 00
(5.48)
83
Using the matter Lagrangian (A.25) as well as the metric ansatz (5.1), and replacing the
gauge fields by their charges, as in (5.10), gives
�
� �
��
¯ J� + g 2 e−2 U V (X̃, X̃)
¯
2ψ
�
�
e2ψ U �2 − ψ �2 + NIJ X̃ I� X̃
+
e
(2ψ
−
2
U
)
= 0 ,(5.49)
tot
where Vtot denotes the combined potential
¯ = g 2 V (X̃, X̃)
¯ + e−4A V (X̃, X̃)
¯
Vtot (X̃, X̃)
BH
�
�
�
�
¯ − 2|W̃ |2 + e−4A N IJ ∂ Z̃∂ Z̃¯ + 2|Z̃|2 . (5.50)
= g 2 N IJ ∂I W̃ ∂J¯W̃
I
J¯
The Hamiltonian constraint (5.49) can now be rewritten (up to a total derivative term) as
LBPS + Lsympl = 0 ,
(5.51)
where LBPS and Lsympl denote the integrands of (5.19) (with the charges and fluxes replaced
by their hatted counterparts) and (5.31), respectively. Since LBPS = 0 on a solution to
the flow equations, it follows that the Hamiltonian constraint reduces to the symplectic
constraint (5.32). The total derivative term vanishes by virtue of the flow equation for
A = ψ − U.
In the following, we briefly check that (5.36) reproduces the standard flow equations
for supersymmetric domain wall solutions in AdS4 . Setting QI = P I = 0, and choosing
the phase γ = π/2, we obtain ψ = 2U , A = U as well as
� I
�
� I�
(Y − Ȳ I )�
h
= ig
,
(5.52)
�
(FI − F̄I )
hI
which describes the supersymmetric domain wall solution of [127]. Observe that the choice
γ = 0 leads to a solution of the type (5.52) with the imaginary part of (Y I , FI )� replaced
by the real part. The flow equations (5.52) can be easily integrated and yield
� I
�
� I�
� I
�
Y − Ȳ I
H
α + hI r
= ig
= ig
,
(5.53)
FI − F̄I
HI
βI + hI r
where (αI , β I ) denote integration constants. This solution satisfies W (Y ) = W̄ (Ȳ ), which
is�precisely the condition
(5.39). In addition, making use of (5.5), we obtain e2U =
�
I
I
g H FI (Y ) − HI Y .
5.1.2
Generalizations of the first-order rewriting
In the preceding section the first-order rewriting was done for a general prepotential. In
minimal gauged supergravity other first-order rewritings are possible. For a detailed discussion see appendix B. Moreover, in the first-order rewriting performed above, what was
used operationally was the fact that i) for an arbitrary charge and flux configuration the
one-dimensional bulk Lagrangian can be written as a sum of squares and ii) imposing
84
the symplectic constraint (5.32), the vanishing of the squares gives a vanishing bulk Lagrangian, which in turn is equal to the Hamiltonian density. This ensures that both the
equations of motion as well as the Hamiltonian constraint are satisfied. Given this procedure, we may identify invariances of the action under transformations of the charges
and/or fluxes, since any such transformation will automatically produce a new, possibly
physically distinct, rewriting of the action.
In the ungauged case in four dimensions this has been already explored in terms of an Smatrix that operates on the charges, while keeping the action invariant [39]1 . In the gauged
case, the presence of both charges and fluxes allows for a wider class of transformations,
in which both sets of quantum numbers are transformed. Thus finding transformations on
both charges and fluxes that leave the action invariant allows for more general rewritings.
In [1] such transformations for the charges and fluxes are constructed. Also some explicit
examples are given. Here we will not go into the details and refer the reader to [1].
Furthermore, in [1] a class of non-extremal black brane solutions that are based on firstorder flow equations is constructed. This is done by turning on a deformation parameter
in the line element describing extremal black branes. In the context of N = 2 U (1) gauged
supergravity in five dimensions, it was shown in [12] that there exists a class of non-extremal
charged black hole solutions that are based on first-order flow equations, even though their
extremal limit is singular. There the non-extremal parameter is not only encoded in the
line element, but also in the definition of the physical charges. In [1] only that case is
studied where the deformation parameter enters in the line element. A different procedure
for constructing non-extremal black hole solutions has recently been given in [128] in the
context of ungauged supergravity in four dimensions. With the rewriting obtained in [1]
it is possible to reproduce e.g. the non-extremal versions of some of our solutions, such as
the solution that will be presented in (5.95). This includes the solution discussed in [9].
Moreover, the non-extremal deformation of the extremal domain wall solution (5.53) and
the non-extremal version of the interpolating solution near AdS4 and near AdS2 × R2 to be
discussed in section 5.2.1 can be reproduced. Again we refer the reader to [1] for further
information.
5.1.3
AdS2 × R2 backgrounds
In the following, we will consider space-times of the type AdS2 × R2 , i.e. line elements (5.1)
with constant A. In view of the relation (5.5), we thus demand that the Y I are constant
in this geometry. Observe that the latter differs from the case of black holes in ungauged
supergravity. There, the appropriate Y I variable is not given in terms of (5.3), but rather
in terms of A = −U , and the associated flow equations are solved in terms of Y I = Y I (r)
rather than in terms of constant Y I .
For constant Y I , their flow equation yields qI = 0, which implies Q̂I = i g e2A ĥI .
Inserting this into the flow equation for U gives
�
�
eA (eU )� = −2g Im eiγ Y I ĥI ,
(5.54)
1
See also [63] for related transformations in five dimensions.
85
which equals (eψ )� . Hence we obtain ψ � = U � , which is consistent with A = ψ − U =
constant. Combining the flow equations (5.45) and (5.54) gives
�
��
eA eU −iγ = 2 i g Y I ĥI = constant ,
(5.55)
which yields
eA eU −iγ = 2 i g Y I ĥI r + c ,
It follows that
c∈C.
(5.56)
�
�
eA eU = Re 2 i g eiγ Y I ĥI r + c eiγ
�
�
�
�
= −2 g Im eiγ Y I ĥI r + Re c eiγ .
(5.57)
Now recall that γ is given by (5.42), which takes a constant value, since the Y I are constant.
Hence Re [c eiγ ] is constant, and it can be removed by a redefinition of r, resulting in
�
�
eA+U = −2 g Im eiγ Y I ĥI r .
(5.58)
�
�
Observe that Im eiγ Y I ĥI �= 0 to ensure that the space-time geometry contains an AdS2
factor.
Contracting Q̂I = i g e2A ĥI with Y I yields the value for e2A as
e2A = −i
Y I Q̂I
J
g Y ĥJ
= −i
and hence
Q̂I =
Z(Y )
Z̄(Ȳ )
=i
,
g W (Y )
g W̄ (Ȳ )
(5.59)
Z(Y )
ĥI .
W (Y )
(5.60)
The values of the Y I are, in principle, obtained by solving (5.60), or equivalently,
QI − 12 (FIJ + F̄IJ )P J =
− 12 NIJ P J =
1
g e2A NIJ hJ ,
2
�
g e2A hI − 12 (FIJ
+ F̄IJ )hJ
�
.
(5.61)
There may, however, be flat directions in which case some of the Y I remain unspecified,
and an example thereof is given in section 5.2. The reality of e2A forces the phases of Z(Y )
and of W (Y ) to differ by π/2 [119].
This relation (5.60) has an immediate consequence, similar to the one for black holes
derived in [119]. Namely, using (5.60) and (5.59) in (5.33) leads to
�
�
�
�
Z(Y )
¯
¯
¯
0 = Im
ĥI N IJ ĥJ = Im i g e2A ĥI N IJ ĥJ = g e2A ĥI N IJ ĥJ .
(5.62)
W (Y )
Given that g e2A �= 0, we infer that for any AdS2 × R2 geometry
¯
ĥI N IJ ĥJ = 0 .
We will use this fact below in section 5.2.1.
(5.63)
86
5.2
Exact solutions
In the following, we consider exact (dyonic) solutions of the flow equations in specific
models. In general, solutions fall into two classes, namely solutions with constant γ and
solutions with non-constant γ along the flow.
5.2.1
Solutions with constant γ
Let us discuss dyonic solutions in the presence of fluxes. For concreteness, we choose γ = 0
in the following. The flow equations (5.34) for U, ψ and A read,
�
�
�
�
(eU )� = e−3A Re Y I Q̂I − g e−A Im Y I ĥI ,
�
�
(eψ )� = −2 g Im Y I ĥI ,
�
�
(eA )� = − Re Y I qI ,
(5.64)
while the flow equations for the Y I are
� I
�
�
�
�
�
��
(Y − Ȳ I )�
N IK Q̂K
N IK ĥK
−ψ
2A
= 2i e
− Im
+ g e Re
.
(FI − F̄I )�
F̄IK N KJ Q̂J
F̄IK N KJ ĥJ
(5.65)
In the presence of both fluxes and charges, the flow equations (5.65) cannot be easily
integrated. A simplification occurs whenever Re FIJ = 0. This is, for instance, the case in
the model F = −iX 0 X 1 , to which we now turn.
The F (X) = −iX 0 X 1 model
For this choice of prepotential, we have F (X) = (X 0 )2 F(z), where F(z) = −iz with
z = X 1 /X 0 = Y 1 /Y 0 . The associated Kähler potential reads K = − ln(z + z̄), and z is
related to the dilaton through Re z = e−2φ . We also have
�
�
�
�
0 1
0 1
IJ
1
NIJ = −2
, N = −2
,
(5.66)
1 0
1 0
as well as
N00 = −iz ,
N11 = −i/z ,
N01 = 0 .
(5.67)
The flow equations (5.65) become

 


ĥ1
Q̂1
(Y 0 − Ȳ 0 )�


 
 (Y 1 − Ȳ 1 )� 
Q̂
ĥ0 



0
−ψ


 − i g e−ψ+2A Re   .
−i(Y 1 + Ȳ 1 )�  = i e Im 
i Q̂0 
i ĥ0 
−i(Y 0 + Ȳ 0 )�
i Q̂1
i ĥ1

(5.68)
5.2 Exact solutions
87
This yields


 0

(Y 0 − Ȳ 0 )�
P − g e2A h1
2A
 (Y 1 − Ȳ 1 )� 
 1


 = i e−ψ Re P − g e2A h10  .
1
1
�
−i(Y + Ȳ ) 
 Q0 + g e h 
−i(Y 0 + Ȳ 0 )�
Q1 + g e2A h0
(5.69)
In order to gain some intuition for finding a solution with all charges and fluxes turned
on, let us first consider a simpler example. We retain only one of the four charge/flux
combinations appearing in (5.69), namely the one with Q1 �= 0, h0 �= 0. Observe that any
of these four combinations satisfies the symplectic constraint (5.32). The associated flow
equations (5.69) are




(Y 0 − Ȳ 0 )�
0
 (Y 1 − Ȳ 1 )� 


0

 = i e−ψ 
 ,
(5.70)
1
1
�
−i(Y + Ȳ ) 


0
−i(Y 0 + Ȳ 0 )�
Q1 + g e2A h0
which yields Y 1 = constant and
�
�
(Y 0 )� = − 12 e−ψ Q1 + g e2A h0 .
(5.71)
The flow equations for ψ and A read,
� �
(eψ )� = −2g Re Y 1 h0 ,
� 2A ��
� ��
�
e
= −2e−ψ Re Y 1 Q1 + g e2A h0 .
(5.72)
�
�
For the equation for A we used q0 = 0 and q1 = e−A−ψ Q1 + g e2A h0 . The flow equation
for ψ can be readily integrated and yields
� �
eψ = −2g Re Y 1 h0 (r + c) ,
(5.73)
where c denotes an integration constant. Inserting this into the flow equation for A gives
which can be integrated to
�
e2A
��
e2A =
=
Q1 + g e2A h0
,
g h0 (r + c)
eβ (r + c) − Q1
,
g h0
(5.74)
(5.75)
where β denotes another integration constant. Plugging this into the flow equation for Y 0 ,
we can easily integrate the latter,
Y0 =
eβ
(r + δ) ,
4gh0 (Re Y 1 )
(5.76)
88
where δ denotes a third integration constant. Using (5.5), we infer
δ = c − e−β Q1 .
(5.77)
Moreover, for U we obtain
2
e
2U
2ψ−2A
=e
4g 3 (h0 )3 (Re Y 1 ) (r + c)2
=
.
eβ (r + c) − Q1
(5.78)
We take the horizon to be at r + c = 0, i.e. we set c = 0 in the following. Summarizing,
we thus obtain
Y 1 = constant ,
eβ r − Q 1
Y0 =
,
4 gh0 (Re Y 1 )
4 g h0 (Re Y 1 )2
Re z =
,
eβ r − Q 1
eβ r − Q 1
4 (Re Y 1 )2
e2A =
=
,
g h0
Re z
2
4g 3 (h0 )3 (Re Y 1 ) r2
e2U =
,
eβ r − Q 1
�
�2
e2ψ = 4g 2 Re Y 1 (h0 )2 r2 .
(5.79)
We require h0 > 0 and Q1 < 0 to ensure positivity of Re z, e2A and e2U . This choice is also
necessary in order to avoid a singularity of Re z and e2U at r = e−β Q1 . The resulting brane
solution has non-vanishing entropy density, i.e. e2A(r=0) �= 0. The imaginary part of z is
left unspecified by the flow equations. However, demanding the constraint (5.41) imposes
Y 1 to be real, so that Im z = 0. Note that Re Y 1 corresponds to a flat direction (see also
[129]). The above describes the extremal limit of the solution discussed in sec. 7 of [9] in
the context of AdS/CMT (see also [130, 131, 132]).2
We now want to solve the equations (5.69) when all charges and fluxes are non-zero.
Since the equations are quite difficult to solve directly, we will make an ansatz for eψ and
e2A and then solve for the Y I . In the example above we saw that eψ and e2A were linear
functions of r, so we choose the following ansatz for these functions,
eψ(r) = a r ,
e2A(r) = b r + c .
When plugging this ansatz into the flow equations (5.69) we get
� gb
1 �¯
¯
¯
(Y 0 (r))� = −
Q̂1 + igcĥ1 − iĥ1 ,
2ar
2a
� gb
1 �¯
¯
¯
(Y 1 (r))� = −
Q̂0 + igcĥ0 − iĥ0 .
2ar
2a
2
(5.80)
(5.81)
In order to compare the solutions, one would have to set γ = δ = 1 in (7.1) of [9], take their solution
2
1
to the extremal limit m2 = q8 and relate the radial coordinates according to r(us) = 2l
(r(them) )2 − 12 √q8 .
5.2 Exact solutions
89
These equations can easily be integrated to give
�
1 �¯
¯
Y 0 (r) = −
Q̂1 + igcĥ1 ln r −
2a
�
1 �¯
¯
Y 1 (r) = −
Q̂0 + igcĥ0 ln r −
2a
gb ¯
iĥ1 r + C 0 ,
2a
gb ¯
iĥ0 r + C 1 .
2a
(5.82)
In order for (5.80) and (5.82) to constitute a solution, several conditions on the parameters
a, b, c, the charges and the fluxes have to be fulfilled. On the one hand, one has to impose
the constraints
�
�
�
�
Im Q̂I Y I = 0 , Re ĥI Y I = 0 ,
(5.83)
following from (5.34) and (5.39) for γ = 0. On the other hand, further constraints on
the parameters arise from (5.5), (5.33) and the equation for ψ in (5.64). Note that the
equation for U does not give additional information, as U is determined, once the Y I and
ψ are given.
First, the constraints (5.83) imply
�
�
¯
Re Q̂I N IJ ĥJ
= 0,
�
�
Im Q̂I C I
= 0,
�
�
Re ĥI C I
= 0.
(5.84)
Next, let us have a look at the equation for ψ. Using our ansatz for eψ , it reads
�
�
a = −2 g Im Y I ĥI .
(5.85)
With the form of the Y I given in (5.82) and demanding that the right hand side of (5.85)
is a constant, we obtain the constraint
�
�
¯
Re ĥ1 ĥ0 = 0 .
(5.86)
This directly leads to a vanishing of the term linear in r. Together with the symplectic
constraint (5.33) it also implies that the logarithmic term vanishes. Moreover, we read off
�
�
a = −2g Im C I ĥI .
(5.87)
Next, we analyze the constraints coming from e2A = −NIJ Y I Ȳ J . With our ansatz
(5.80) and the form of NIJ given in (5.66), one obtains
�
�
b r + c = 4 Re Y 0 Ȳ 1 .
(5.88)
This fixes the constant c to be
c = 4 Re(C 0 C̄ 1 ) .
(5.89)
90
Using (5.87) the term linear in r on the right hand side of (5.88) is just identically b r,
i.e. there is no constraint on b (except that it has to be positive in order to guarantee the
positivity of e2A for all r). All other terms on the right hand side vanish if one demands
Re
��
¯
Q̂I N IJ Q̂J = 0 ,
� �
Q̂I − igcĥI C I = 0 ,
(5.90)
(5.91)
in addition to (5.86) and the symplectic constraint (5.33).
We now show that (5.91) is fulfilled because the even stronger constraint
Q̂I − igcĥI = 0
(5.92)
holds. To do so, we note that (5.91) together with (5.84) and (5.87) imply
ac
,
2
a
= −i .
2g
Q̂0 C 0 + Q̂1 C 1 = −
ĥ0 C 0 + ĥ1 C 1
(5.93)
To get a solution for C 0 and C 1 one of the two following conditions has to be valid:
i) If and only if Q̂0 ĥ1 − ĥ0 Q̂1 = ρ �= 0 there is a unique solution for C 0 and C 1 .
ii) If the two lines in (5.93) are multiples of each other then there exists a whole family
of solutions.
We will now show that case i) can be ruled out. To do so we combine the determinant
condition of case i) with the first constraint in (5.84), i.e. we look at the system of linear
equations for Q̂0 and Q̂1
Q̂0 ĥ1 − Q̂1 ĥ0 = ρ ,
¯
¯
Q̂0 ĥ1 + Q̂1 ĥ0 = 0 .
(5.94)
Obviously, the two lines cannot be multiples of each other for ρ �= 0. Thus, in order
¯
¯
to find a solution at all, the determinant ĥ1 ĥ0 + ĥ1 ĥ0 has to be non-vanishing. This is,
however, in conflict with (5.86). Thus, the two lines in (5.93) have to be multiples of each
other, implying (5.92). Note that this implies the absence of the logarithmic terms in the
solutions for Y I .
To summarize we have found the following solution:
gb ¯
iĥ1 r + C 0 ,
2a
gb ¯
Y 1 (r) = − iĥ0 r + C 1 ,
2a
eψ(r) = ar ,
e2A(r) = br + c .
Y 0 (r) = −
(5.95)
5.2 Exact solutions
91
In addition, the parameters have to fulfill the conditions
¯
Re(ĥ0 ĥ1 ) = 0 ,
Q̂I = igcĥI ,
ĥI C I = −i
a
,
2g
Re(C 0 C̄ 1 ) =
c
,
4
(5.96)
while the parameter b can be any non-negative number. For b = 0 this solution falls into
the class discussed in section 5.1.3.
Let us finally also mention that one can show that the F (X) = −iX 0 X 1 model does
not allow for Nernst brane solutions (i.e. solutions with vanishing entropy) of the firstorder equations. Making an ansatz eU ∼ rα , eψ ∼ rβ and eA ∼ rβ − α for the near-horizon
geometry, one can show that the only solutions with both non-vanishing charges and fluxes
have α = β = 1 and are captured by the solution discussed in this section after setting
b = 0.
The F = − (X 1 )3 /X 0 model: Interpolating solution between AdS4 and AdS2 × R2
Next, we would like to construct a (supersymmetric) solution that interpolates between
an AdS4 vacuum at spatial infinity, and an AdS2 × R2 background with constant Y I at
r = 0. Thus, asymptotically, we require the solution to be of the domain wall type (see
(5.52)) with γ = π/2. Hence the Y I satisfy (5.53) with αI = βI = 0, so that we may write
¯
I
I
Y I = y∞
r with y∞
= g N IJ ĥJ = constant. The latter can be established as follows. Using
I
I
(Y − Ȳ ) = ig h r and FI − F̄I = ig hI r, we obtain
¯
iNIJ Y J = FI − F̄IJ Y J = ig ĥI r ,
so that
1
(Y
2
It follows that asymptotically,
�
�
¯
+ Ȳ )I = g N IJ ĥJ − 12 i hI r .
(5.97)
(5.98)
¯
Y I = g N IJ ĥJ r ,
(5.99)
¯
e2A = e2U = −g 2 ĥI N IJ ĥJ r2 .
(5.100)
and hence, in an AdS4 background,
Since this expression only depends on the Y I through the combinations NIJ and FIJ , which
are homogeneous of degree zero, the r-dependence scales out of these quantities, and thus
¯
¯
ĥI N IJ ĥJ is a constant. For e2A to be positive, we need to require a2 ≡ −ĥI N IJ ĥJ > 0.
At r = 0, on the other hand, we demand γ = γ0 as well as Y I = constant, with A given
by (5.59). Thus, we want to construct a solution with a varying γ(r) that interpolates
¯
between the values π/2 and γ0 . From (5.63) we know that ĥI N IJ ĥJ = 0 at the horizon.
The Y I appearing in this expression are evaluated at the horizon. In general, their values
will differ from the asymptotic values, so that the flow will interpolate between an asymp¯
totic AdS4 background satisfying ĥI N IJ ĥJ < 0 and an AdS2 × R2 background satisfying
92
¯
ĥI N IJ ĥJ = 0. This, however, will not be possible whenever FIJ is independent of the Y I ,
such as in the F = −iX 0 X 1 model, as already observed in [119]. Thus, in the example
below, we will consider the F = − (X 1 )3 /X 0 model instead.
The interpolating solution has to have the following properties. At spatial infinity,
where Im W (Y ) = 0, γ is driven away from its value π/2 by the term Re Z(Y ) in the flow
equation of γ,
�
�
γ = e
and hence
2U −3ψ
Re Z(Y )
�
∞
�
= e
−4U
Re Z(Y )
�
∞
=
Re Z(y∞ )
,
a4 r 3
(5.101)
π Re Z(y∞ )
−
.
(5.102)
2
2a4 r2
Our example below has Re Z(y∞ ) = 0 though and, thus, the phase γ will turn out to be
constant. Near r = 0, on the other hand, the deviation from the horizon values can be
determined as follows. Denoting the deviation by δY I = β I rp , we obtain δ(e2A ) = c rp
with
c = −NIJ (β I Ȳ J + Y I β̄ J ) ,
(5.103)
γ(r) ≈
where in this expression the Y I and NIJ are calculated at the horizon. Using this, we
compute the deviation of q̄I from its horizon value q̄I = 0,
�
�
¯
δ q̄I = e−A−ψ−iγ0 −F̄I¯J¯L̄ (pJ + ighJ )β̄ L + ig ĥI c rp ,
(5.104)
where all the quantities that do not involve β I are evaluated on the horizon. The flow
equations for the Y I then yield
p NIK β K =
where
�
e−iγ0 �
¯
−F̄I¯J¯L̄ (pJ + ighJ )β̄ L + ig ĥI c ,
∆
(5.105)
�
�
∆ = −2g Im eiγ0 ĥI Y I .
(5.106)
Contracting (5.105) with Ȳ I yields p = 1. Inserting this value into (5.105) yields a set of
equations that determines the values of the β I .
Next, using the flow equation for γ, we compute the deviation from the horizon value
γ0 , which we denote by δγ = Σ r. We obtain
�
�
g
Σ = − Re eiγ0 β I ĥI .
(5.107)
∆
The example below will have a vanishing Σ, consistent with a constant γ. Finally, using
the flow equation for ψ we compute the change of ψ,
�
�
g
δψ = − Im eiγ0 β I ĥI r .
(5.108)
∆
5.2 Exact solutions
93
We now turn to a concrete example of an interpolating solution. As already mentioned,
1 3
this will be done in the context of the F (X) = − (XX 0) model. To obtain an interpolating
solution we first need to specify the form of the solution at both ends.
Let us first have a look at the near horizon AdS2 × R2 region. According to our
discussion in sec. 5.1.3, we have to solve (5.63) and Q̂I = ige2A ĥI under the assumption
that the Y I are constant. It turns out that these constraints can be solved, for instance,
by choosing Q0 , P 1 , h1 , h0 �= 0 and all other parameters vanishing. Of course, due to the
symplectic constraint (5.32), the four non-vanishing parameters are not all independent,
but have to fulfill
Q0 h 0
P1 =
.
(5.109)
h1
1 3
1
For F (Y ) = − (YY 0) , and introducing z = z1 + iz2 = YY 0 , we have
�
�
−4 Im (z 3 ) 6 Im (z 2 )
NIJ =
.
6 Im (z 2 ) −12 Im (z)
(5.110)
Using
e2A = 8|Y 0 |2 z23 ,
(5.111)
which follows from (5.5), shows that one can fulfill Q̂I = ige2A ĥI and (5.63) by fixing
z1 = 0 ,
�
√
(3 + 2 3)h1
z2 =
,
3h0
1√
3 4 Q0
0
|Y | = √ 2 √ .
2 2z2 h1
(5.112)
Interestingly we find that the axion, which is in this model given by z1 , has to vanish and
all parameters Q0 , P 1 , h1 , h0 have to have the same sign.
In order to describe the asymptotic AdS4 region, we note that asymptotically the
charges Q0 and P 1 can be neglected and we can read off the asymptotic form of the
solution from (5.53) with vanishing integration constants αI and βI . More precisely, we
have to allow that the asymptotic form of the interpolating solution differs from (5.53) by
an overall factor. This is because, apriori, we only know that asymptotically (ψ − 2U )� = 0,
cf. (5.34) for vanishing charges. Without the need to match the asymptotic region to a near
horizon AdS2 × R2 region, we could just absorb the constant ψ − 2U in a rescaling of the
coordinates x and y. However, when we start from the hear horizon solution and integrate
out to infinity, we can not expect to end up with this choice of convention. Allowing for a
non-vanishing constant ψ − 2U = C �= 0, the flow equations (5.52) for Y I and, thus, also
the solutions (5.53), would obtain an overall factor eC . Concretely, we obtain
g
0
YAdS
(r) = ieC h0 r ,
4
2 √
0
1
C g h h1
√
YAdS
(r)
=
−e
r,
(5.113)
4
2 3
94
C
with a constant
� e to be determined by numerics below. Note, however, that the dilaton
h1
0
e−2φ = z2 = 3h
0 is independent of this factor and it comes out to be real if h1 and h
have the same sign, consistently with what we found from the near horizon region.
We would now like to discuss the interpolating solution, which we obtain by specifying
boundary conditions at the horizon and then integrating numerically from the horizon
to infinity using Mathematica’s NDSolve. Given that in the AdS4 region Y 0 is purely
imaginary and Y 1 is purely real, we choose the same reality properties at the horizon, i.e.
we fulfill (5.112) by
Yh0
Yh1
√
5
3 4 h0 Q0
= i √
3 ,
√
2 2(3 + 2 3)h12
3�
3 4 h0 Q0
0
= iYh z2 = − �
.
√
2 2(3 + 2 3)h1
(5.114)
The subscript h means that these values pertain to the horizon. Now we can determine
the phase γ0 at the horizon using (5.42). We get e−2iγ0 = −1 independently of h1 , h0 , Q0
and P 1 . We choose γ0 = π/2 as in the asymptotic AdS4 region and we will see that this
leads to a constant value for γ throughout.
Next we have to determine how the solution deviates near the horizon from AdS2 × R2 .
We do this following our discussion at the beginning of this section, cf. (5.103) - (5.108).
Given the reality properties of Y I in both limiting regions, we choose β 0 purely imaginary
and β 1 purely real. Solving (5.105) for the β’s results in
�
√ �
√
�√
�
h0 ( 3 − 2) 3 + 2 3h1 hh10 + (5 3 − 9)Q0 )
Im β 0 = Re β 1
.
(5.115)
h21
Given this, the deviation of all the fields close to the horizon can be determined (resulting
in initial conditions at, say ri = 10−6 ) and a numerical solution for the flow equations
(5.34) can be found, integrating from the horizon outwards. To do the numerics we chose
g=1 ,
Re β 1 = 0.01 ,
Q0 = 1 ,
h1 = 1 ,
h0 = 10 .
(5.116)
One can see that asymptotically Im Y 0 and Re Y 1 are linear functions of r (cf. fig. 5.1),
whereas the real part of Y 0 and imaginary part of Y 1 are zero within the numerical tolerances. The constant eC in (5.113) can be determined to be roughly eC ≈ 422. Futhermore,
for r → ∞, the functions eA , eU and eψ behave as expected for the AdS4 background, i.e.
eA = O(r), eU = O(r) and eψ = O(r2 ) (cf. fig. 5.2 and 5.3).3 Another important feature is
that γ(r) = π2 for any r, cf. fig. 5.3.
3
Note the difference in the plotted r-range in fig. 5.3 compared to figs. 5.1 and 5.2. Plotting Im Y 0 ,
Re Y 1 , eA and eU for larger values would just confirm the linearity in r.
5.2 Exact solutions
95
1
0
Re Y (r)
Im Y (r)
r
2.168
1.5 ! 10"6 2. ! 10"6 2.5 ! 10"6 3. ! 10"6 3.5 ! 10"6 4. ! 10"6
2.166
"1.0025
2.164
"1.0030
2.162
2.160
"1.0035
r
2. ! 10"6 3. ! 10"6 4. ! 10"6 5. ! 10"6 6. ! 10"6 7. ! 10"6 8. ! 10"6
"1.0040
Figure 5.1: Interpolating solutions for Re Y 1 and Im Y 0 .
eA(r)
eU (r)
3.9 ! 10"6
1.936
3.8 ! 10"6
3.7 ! 10"6
1.934
3.6 ! 10"6
3.5 ! 10"6
1.932
3.4 ! 10"6
3.3 ! 10"6
2. ! 10"6
4. ! 10"6
6. ! 10"6
8. ! 10"6
0.00001
r
r
1.05 ! 10"6
1.1 ! 10"6
1.15 ! 10"6
1.2 ! 10"6
Figure 5.2: Interpolating solutions for eA and eU .
5.2.2
Solutions with non-constant γ
Next, we turn to solutions with non-constant γ along the flow. We focus on two models.
The F = − i X 0 X 1 model
We first consider the model F = − i X 0 X 1 and restrict ourselves to a case involving only
the electric charge Q0 and the electric flux h0 . Since N IJ is off-diagonal, the flow equation
for Y 0 reads (Y 0 )� = 0. This gives Y 0 = C 0 , with C 0 a non-vanishing c-number, which
we take to be real, since the
of C 0 can be absorbed into γ. Then, the relation (5.5)
� 1phase
�
0
yields exp(2A) = 4C Re Ȳ , from which we infer that Re Y 1 = exp(2A)/(4C 0 ). In the
following we take Q0 , h0 > 0 and C 0 < 0, for concreteness.
The constraint equation (5.39) yields the following relation between γ and A,
tan [ γ ] =
g h0 2A
e .
Q0
(5.117)
Using the τ -variable introduced in (5.47), the flow equation for A can be written as
�
�
Ȧ = − Re ( Q0 e− 2A − i g h0 ) C 0 eiγ
�
�
= − C 0 Q0 e−2A cos [ γ ] + g h0 sin [ γ ] ,
(5.118)
96
e
ψ(r)
γ(r)
3.0
5 ! 1014
2.5
4 ! 1014
2.0
3 ! 1014
1.5
2 ! 1014
1.0
14
1 ! 10
0.5
200 000
400 000
600 000
r
800 000
200 000
1 ! 106
400 000
600 000
800 000
r
1 ! 106
Figure 5.3: Interpolating solutions for eψ and γ.
where Ȧ = ∂A/∂τ . Using (5.117) and (assuming γ ∈ [−π/2, π/2])
tan [ γ ]
sin [ γ ] = �
1 + tan2 [ γ ]
in (5.118) gives
∂ 2A
e = −2 C 0
∂τ
This is solved by
e
2A
�
= α e
δτ
�
(5.119)
Q20 + g 2 h20 e4A .
Q20
−
e−δ τ
4 α2 g 2 h20
�
,
(5.120)
(5.121)
where δ = −2 g h0 C 0 > 0 and α denotes an integration constant. Setting α = Q0 /(2gh0 )
this becomes
Q0
e2A =
sinh( δ τ ) ,
(5.122)
gh0
and demanding exp(2A) to be positive restricts the range of τ to lie between 0 and ∞.
The flow equation for ψ is given by
ψ̇ = δ sin [ γ ] = δ tanh( δ τ ) ,
(5.123)
where we used (5.119), (5.117) and (5.122). This can easily be solved by
eψ−ψ0 = cosh( δ τ ) ,
(5.124)
where ψ0 denotes an integration constant which we set to zero. Using dr = exp(ψ) dτ , we
establish
sinh( δ τ )
r − r0 =
>0.
(5.125)
δ
As τ → 0 we have exp(2A) → 0 and r − r0 → 0.
The dilaton is given by
� 1�
Y
e2A
Re z = Re
=
.
(5.126)
Y0
4 (C 0 )2
5.2 Exact solutions
97
√
We notice here that the dilaton eφ = ( Re z)−1 blows up at r → r0 . The same happens
with the curvature invariants like the Ricci scalar. As the curvature invariants vanish
asymptotically for large r (we checked this up to second order in the Riemann tensor), one
could still hope that our solution describes the asymptotic region of a global solution, once
higher curvature corrections are taken into account.
The STU model
Now we consider the STU-model, which is based on F (Y ) = −Y 1 Y 2 Y 3 /Y 0 . We denote the
z i = Y i /Y 0 (with i = 1, 2, 3) by z 1 = S , z 2 = T , z 3 = U . For concreteness, we will only
consider solutions that are supported by an electric charge Q0 and an electric flux h0 . In
addition, we will restrict ourselves to axion-free solutions, that is solutions with vanishing
Re S , Re T , Re U . The solutions will thus be supported by S2 = Im S , T2 = Im T , U2 =
Im U , so that


4S2 T2 U2
0
0
0

0
0
−2U2 −2T2 
,
NIJ = 

0
−2U2
0
−2S2 
0
−2T2 −2S2
0


1
0
0
0
0
1
S22
−S2 T2 −S2 U2 

.
N IJ =
(5.127)

T22
−T2 U2 
4S2 T2 U2 0 −S2 T2
0 −S2 U2 −T2 U2
U22
The flow equations for the Y i imply that they are constant. To ensure that S, T and U
are axion-free, we take the constant Y i to be purely imaginary, i.e. Y i = i C i , where the
C i denote real constants. Using (5.5) in the form
e2A = 8 |Y 0 |2 S2 T2 U2 ,
(5.128)
the flow equation (5.47) for Y 0 gives
Ẏ 0 = 2 |Y 0 |2 ( Q0 e− 2 A + i g h0 ) e− i γ .
(5.129)
Using the constraint equation (5.39) in the form
�
�
Im (Q0 e− 2A + i g h0 ) e− iγ Ȳ 0 = 0
(5.130)
as well as the flow equation for A,
Ȧ = − Re
we can rewrite (5.129) as
��
Q0 e− 2A + i g h0
Ẏ 0 = − 2 Y 0 Ȧ .
�
�
e− iγ Ȳ 0 ,
(5.131)
(5.132)
98
This immediately gives
Y 0 = C 0 e− 2A ,
(5.133)
where C 0 denotes an integration constant which we take to be real, for simplicity. Then,
it follows from (5.128) that
C1 C2 C3
8
= 1.
(5.134)
C0
For concreteness we take C I < 0 (I = 0, . . . , 3) in the following (ensuring the positivity of
S2 , T2 and U2 ), as well as h0 , Q0 > 0.
The constraint equation (5.130) yields the following relation between γ and A,
tan [ γ ] =
g h0 2A
e .
Q0
(5.135)
The flow equation (5.131) can be written as
1
2
�
�
∂ 2A
e = − C 0 Q0 e−2A cos [ γ ] + g h0 sin [ γ ]
∂τ
(5.136)
which, using (5.135), leads to
1
4
∂ 4A
e = −C 0
∂τ
This can be solved to give
�
Q20 + g 2 h20 e4A .
(5.137)
2
e4A =
4 g 4 h40 (C 0 ) (τ + c )2 − Q20
,
g 2 h20
(5.138)
where c denotes a further integration constant. Hence, the prefactor of the planar part of
the metric is
�
4 g 4 h40 (C 0 )2 (τ + c )2 − Q20
2A
e =
.
(5.139)
g h0
This is well behaved provided that (τ + c)2 ≥ Q20 /(4g 4 h40 (C 0 )2 ).
The flow equation for ψ,
ψ̇ = −2 g h0 C 0 e− 2A sin [ γ ]
(5.140)
is solved by (using (5.119), (5.135) and (5.138))
eψ =
1
.
τ +c
(5.141)
The radial coordinate is then related to the τ variable by
r [ τ ] = log [ τ + c ] ,
(5.142)
5.3 Nernst brane solutions in the STU model
99
where we set an additional integration constant to zero.
The physical scalars are given by
S2 =
C 1 2A
e
,
C0
T2 =
C 2 2A
e
,
C0
U2 =
C 3 2A
e .
C0
(5.143)
They are positive as long as e2A is. However, as in the previous example, they vanish at the
lower end of the radial coordinate (indicating that string loop and α� corrections should
become important). Again also the curvature blows up there and one can at best consider
this solution as an asymptotic approximation to a full solution which might exist after
including higher derivative terms. As we said in the introduction, it would be worthwhile
to further pursue the search for everywhere well behaved solutions with non-constant γ as
they might be radically different from the ungauged case.
5.3
Nernst brane solutions in the STU model
In the following we construct Nernst brane solutions (i.e. solutions with vanishing entropy
density) in a particular model, namely the STU-model already discussed in sec. 5.2.2. As
there, we denote the z i = Y i /Y 0 (with i = 1, 2, 3) by z 1 = S , z 2 = T , z 3 = U . For
concreteness, we will only consider solutions that are supported by the electric charge
Q0 and the electric fluxes h1 , h2 , h3 . In addition, we will restrict ourselves to axion-free
solutions, that is solutions with vanishing Re S , Re T , Re U . The solutions will thus be
supported by S2 = Im S , T2 = Im T , U2 = Im U , and the corresponding matrices NIJ
and N IJ are given in eq. (5.127). We thus have that e2A is determined by (5.128). In the
following, we take Y 0 to be real, so that the Y i will be purely imaginary.
Let us consider the flow equation (5.47) for the Y I . We set γ = 0. Instead of working
with a τ coordinate defined by dτ = e−ψ dr, we find it convenient to work with dτ =
−e−ψ dr. We obtain
Q0
,
4S2 T2 U2
�
�
2 i g Y i 2 Y i hi − Y j hj ,
Ẏ 0
= −
Ẏ i
=
(5.144)
where i, j = 1, 2, 3 and Ẏ I = ∂Y I /∂τ . Here, i is not being summed over, while j is. The
flow equations for Y i are solved by
Yi =−
i
.
2 g hi τ
(5.145)
Next, using that for an axion-free solution S2 = − i Y 1 /Y 0 , T2 = − i Y 2 /Y 0 and U2 =
− i Y 3 /Y 0 , and inserting (5.145) into the flow equation for Y 0 , we obtain
� �3
Ẏ 0 = 2g 3 Q0 h1 h2 h3 Y 0 τ 3 .
(5.146)
100
This equation can be easily solved to give
1
Y 0 = −�
,
3
−g Q0 h1 h2 h3 ( τ 4 + C 0 )
(5.147)
where C 0 denotes an integration constant. We also take Q0 < 0 and h1 , h2 , h3 > 0. This
ensures that the physical scalars
�
1
S2 =
−g 3 Q0 h1 h2 h3 ( τ 4 + C 0 ) ,
2 g h1 τ
�
1
T2 =
−g 3 Q0 h1 h2 h3 ( τ 4 + C 0 ) ,
2 g h2 τ
�
1
U2 =
−g 3 Q0 h1 h2 h3 ( τ 4 + C 0 )
(5.148)
2 g h3 τ
take positive values.
Next, we consider the flow equation for ψ following from (5.34). Using (5.145), we
obtain
�
�
3
ψ̇ = 2 g Im Y i hi = − ,
(5.149)
τ
which upon integration yields
1
eψ−ψ0 = 3 ,
(5.150)
τ
where ψ0 denotes an integration constant, which we set to ψ0 = 0. We can thus take τ to
range between 0 and ∞. The relation dτ = −e−ψ dr then results in
1
r[τ ] =
+ Cr ,
(5.151)
2 τ2
where Cr denotes a real constant, which we take to be zero in the following. This sets the
range of r to vary from ∞ to 0 as τ varies from 0 to ∞.
Using (5.128), we obtain
√
√
−Q0
τ 4 + C0
2A
e = �
,
(5.152)
τ3
g 3 h1 h2 h3
and hence
√
√
−Q0
�
e
=
τ 4 + C0 τ 3 .
(5.153)
3
g h1 h2 h3
We take C0 to be non-vanishing and positive to ensure that the curvature scalar remains
finite througout and that the above describes a smooth solution of Einstein’s equations.
Asymptotically (τ → 0) we obtain
�
�
1
2A r→∞
e
−→ α 1 +
(2 r)3/2 ,
8 C 0 r2
�
� � �3/2
1
1
r→∞
e−2U −→ α 1 +
,
8 C 0 r2
2r
�
�
1
2U r→∞
−1
e
−→ α
1−
(2 r)3/2 ,
(5.154)
8 C 0 r2
−2U
5.3 Nernst brane solutions in the STU model
101
�
where α =
−Q0 C 0 /(g 3 h1 h2 h3 ). Observe that both exp(2A) and exp(2U ) have an
3/2
asymptotic r
fall-off, which is rather unusual. Finally, observe that the scalar fields
1/2
S2 , T2 and U2 grow as (C 0 r) asymptotically.
The near-horizon solution (which corresponds to τ → ∞) can be obtained by setting
0
C = 0 in the above. This yields
√
−Q0
r→0
2A
e
−→ �
(2 r)1/2 ,
g 3 h1 h2 h3
√
−Q0
1
r→0
−2U
e
−→ �
,
(5.155)
g 3 h1 h2 h3 (2 r)5/2
while S2 , T2 and U2 approach the horizon as r−1/2 . The near-horizon geometry has an
infinitely long radial throat as well as a vanishing area density, which indicates that the
solution describes an extremal black brane with vanishing entropy density. It describes a
supersymmetric Nernst brane solution that is valid in supergravity, and hence in the supergravity approximation to string theory when switching off string theoretic α� -corrections.
When embedding this solution into type IIA string theory, where S, T and U correspond
to Kähler moduli, the fact that asymptotically and at the horizon S2 , T2 and U2 blow up
indicates that this solution should be viewed as a good solution only in ten dimensions. In
the heterotic description, where S2 is related to the inverse of the heterotic string coupling
gS , the behaviour S2 = ∞ (which holds asymptotically and at the horizon) is consistent
with working in the classical limit gS → 0.4 One problem that comes across when studying
these Nernst solutions is that they suffer from divergent tidal forces. These may get cured
by quantum or stringy effects [133].
We observe that the Nernst brane solution constructed above differs substantially from
the one constructed in [122]. This is related to the fact that here we deal with a flux
potential in gauged supergravity, and not with a cosmological constant as in [122].
The solution given above is supported by electric charges and fluxes, only. Additional
supersymmetric Nernst solutions can be generated through the technique of symplectic
transformations.
4
In fact, by analyzing this solution in the STU model viewed as arising from heterotic compactification
on K3 × T 2 , one can explicitly verify that (for appropriately chosen flux parameters) the Nernst solution
is a smooth and consistent solution of the 10D supergravity action arising as the low-energy limit of the
h1
2
heterotic string. In particular, the 10D string coupling is given by g10
s = VK3 h3 and, thus, can be made
small for appropriate choices of h1 and h3 , even for values of the K3 volume VK3 which are large enough
to allow the neglect of α� -corrections.
102
Chapter 6
Extremal black brane solutions in
gauged supergravity in D = 5
In this chapter we study stationary black brane solutions in the context of N = 2, U (1)
gauged supergravity in five dimensions. Using the formalism of first-order flow equations,
we construct examples of extremal black brane solutions including Nernst branes, i.e.
extremal black brane solutions with vanishing entropy density, as well as black branes with
cylindrical horizon topology. The work presented here is based on [2].
Since we could not determine Nernst solutions in four dimensions with AdS4 asymptotics
we turn our attention to the case of five dimensions trying to find a Nernst solution with
the desired asymptotic behavior.
Just as in four dimensions the attractor mechanism for extremal black objects guarantees the existence of first-order flow equations. Various types of extremal five-dimensional
black solutions with flat horizons have already been discussed e.g. in [8, 134, 135, 136, 137,
117, 138, 139, 140, 14, 10, 141].
In the following we proceed by deriving the first-order flow equations. We find that the
solution space of these equations allows for Nernst solutions with constant scalars and AdS5
asymptotics. Moreover, we find solutions with constant scalars that may have magnetic
fields or rotation, but no electric charges. Some of them are extremal BT Z × R2 solutions.
For these solutions we can compute the entropy density using the Cardy formula of the
dual CFT. The angular momentum, electric quantum numbers and magnetic fields are
combined in such a way that the resulting quantitiy is invariant under the spectral flow of
the theory, exactly as in the ungauged case [11]. We also construct solutions with running
scalar fields numerically. These interpolate between a BT Z ×R2 near horizon geometry and
AdS5 . We will also display the general relation between the four-dimensional first-order
equations derived in the last chapter and the five dimensional ones.
Because of the difficulties in finding solutions with electric charge we perform a different
first-order rewriting in appendix E. Here we derive a different set of first-order equations
based on [12]. In this setup we are able to construct solutions with electric charge. We can
also reproduce solutions found in [13, 14, 10].
104
6. Extremal black brane solutions in gauged supergravity in D = 5
But let us start with the derivation of the first of the two sets of first-order flow equations.
6.1
First-order equations for stationary solutions in
five dimensions
In the following, we derive first-order flow equations for extremal stationary black brane
solutions in N = 2, U (1) gauged supergravity in five dimensions with n Abelian vector
multiplets. We work in big moduli space. We follow the exposition given in [142] for the
ungauged case and adapt it to the gauged case.
6.1.1
Flow equations in big moduli space
Following [117], we make the ansatz for the black brane line element,
ds2 = −e2U (r) dt2 + e2V (r) dr2 + e2B(r) (dx2 + dy 2 ) + e2W (r) (dz + C(r)dt)2 ,
(6.1)
while for the Abelian gauge fields AA
M (A = 1, . . . , n) we take
M
A
A
A
A
A
A
AA
= AA
M dx
t dt + P x dy + Az dz = (e + Az C(r)) dt + P x dy + Az dz .
(6.2)
A
Here the P A are constants and AA
t , Az depend only on r. The associated field strength
components read
�
A �
A �
A �
FrtA = (AA
t ) = (e ) + Az C + (Az ) C ,
A
Fxy
= PA ,
A
�
Frz
= (AA
z) ,
(6.3)
where � denotes differentiation with respect to r, and (eA )� corresponds to the four-dimensional electric field upon dimensional reduction. The solutions we seek will be supported
by real scalar fields X A (r) and by electric fluxes hA . The ansatz (6.1) and (6.2) is the
most general ansatz with translational invariance in the coordinates t, x, y and z and with
rotational invariance in the x, y-plane, cf. [141].
The bosonic part of the five-dimensional action describing N = 2, U (1) gauged supergravity is given by [143, 144]
�
�
�
√
1
5
A
B MN
S =
dx
−g R − GAB ∂M X A ∂ M X B − GAB FM
NF
2
�
−g 2 (GAB hA hB − 2(hA X A )2 )
�
1
A
B
C KLM N P
− CABC FKL FM N AP �
,
(6.4)
24
6.1 First-order equations for stationary solutions in five dimensions
105
where the scalar fields X A satisfy the constraint 16 CABC X A X B X C = 1. The target space
metric GAB is given by
1
9
GAB = − CABC X C + XA XB ,
(6.5)
2
2
where
2
1
XA = GAB X B = CABC X B X C .
3
6
Inserting the solution ansatz into this action, we find that the Ricci
�
√
2B+W +U −V
−g R = e
2B �2 + 2U � W � + 4B � W � + 4B � U � +
�
��
− 2e2B+W +U −V (2B � + W � + U � ) ,
(6.6)
scalar contributes
�
1 2W −2U �2
e
C
2
(6.7)
while the gauge field kinetic terms contribute
�
�
�
√
1
A
B MN
2B+W +U −V
−g − GAB FM
F
=
e
− GAB P A P B e2V −4B + GAB FrtA FrtB e−2U
N
2
�
A �
B � −2W
−2U 2
A
B � −2U
−GAB (Az ) (Az ) (e
− e C ) − 2GAB Frt (Az ) e C ,
(6.8)
with FrtA given in (6.3). The Chern-Simons term, on the other hand, can be rewritten as
� �
�
�
�
�
1
5
A
B
C KLM N P
B C
= dx5 −CABC FrtA Fxy
Az + T D , (6.9)
dx − CABC FKL FM N AP �
24
where T D denotes a total derivative term. Inserting these expressions into (6.4) yields the
one-dimensional Lagrangian L,
�
1
2B+W +U −V
L = e
2B �2 + 2U � W � + 4B � W � + 4B � U � + e2W −2U C �2 − GAB (X A )� (X B )�
2
B �2 −2U
− GAB P A P B e2V −4B + GAB (eA )� (eB )� e−2U + GAB AA
z Az C e
� −2U
�
B � −2W
+ 2GAB (eA )� AB
− GAB (AA
z C e
z ) (Az ) e
�
2 2V
AB
A 2
− g e (G hA hB − 2(hA X ) )
A B C �
A � B C
− CABC (eA )� P B AC
z − CABC Az P Az C − CABC (Az ) P Az C ,
(6.10)
where we dropped total derivative terms.
Now we express the electric field (eA )� in terms of electric charges QA by performing
the Legendre transformation LL = L − QA (eA )� , and obtain
1
�
(eA )� = e−2B−W +U +V GAB q̂B − AA
zC ,
2
(6.11)
106
where
q̂A = QA + CABC P B AC
z .
(6.12)
Substituting this relation in (6.10) gives
�
1
2B+W +U −V
L = e
2B �2 + 2U � W � + 4B � W � + 4B � U � + e2W −2U C �2 − GAB (X A )� (X B )�
2
1
− GAB P A P B e2V −4B − GAB q̂A q̂B e−4B−2W +2V
4
�
A �
B � −2W
2 2V
AB
A 2
− GAB (Az ) (Az ) e
− g e (G hA hB − 2(hA X ) )
� B C
A �
− CABC (AA
z ) P Az C + QA Az C .
(6.13)
Furthermore, using
B C
�
A � B C
A B C �
(CABC AA
z P Az C) = 2CABC (Az ) P Az C + CABC Az P Az C ,
(6.14)
we obtain
1
� B C
A B C �
− CABC (AA
(6.15)
z ) P Az C = CABC Az P Az C + T D ,
2
where T D denotes again a total derivative, which we drop in the following.
Next, we express C � in terms of a constant quantity J which, in the compact case,
corresponds to angular momentum. We do this by performing the Legendre transformation
LL = L − JC � , and obtain
C � = e−2B−3W +U +V Jˆ ,
(6.16)
where
1
A B C
Jˆ = J − QA AA
z − CABC Az Az P .
2
(6.17)
This results in
L = e
2B+W +U −V
�
2B �2 + 2U � W � + 4B � W � + 4B � U � − GAB (X A )� (X B )�
1
�
B � −2W
−GAB (AA
− GAB P A P B e2V −4B − GAB q̂A q̂B e−4B−2W +2V
z ) (Az ) e
4
�
1 −4B−4W +2V ˆ2
2 2V
AB
A 2
− e
J − g e (G hA hB − 2(hA X ) ) .
(6.18)
2
Now we rewrite the one-dimensional Lagrangian (6.18) as a sum of squares of first-order
flow equations. To this end, we use the relation
�
��
�
� �
��
1
A B C
U −W
�
�
A �
ˆ
eU −W (q̂A AA
−
C
A
P
A
)
=
e
(U
−
W
)(−
J)
+
q̂
(A
)
+ JeU −W ,
ABC z
A
z
z
z
2
(6.19)
and obtain
107
�
�
��
�
1 AC
1 BD
�
−2B+V
B �
−2B+V
− e−2W GAB (AA
)
+
G
q̂
e
(A
)
+
G
q̂
e
C
D
z
z
2
2
�2
1 � ˆ −2B−2W +V
−
Je
− (U � − W � )
2
�
�2
1 �
�
�
A V −2B
3
− B − (U + W ) + 2 XA P e
2
�
�2
�
�
1
3
�
�
�
A
V
A V −2B
1
+
3 B + 2 (U + W ) − 2gX hA e + XA P e
3
2
�
�
��
2
�A
V
C
D −2B
A
AC
D −2B
−GAB X − e
X (ghC + GCD P e
)X − G (ghC + GCD P e
)
3
�
�
��
�B
V 2 E
F −2B
B
BE
F −2B
X −e
X (ghE + GEF P e
)X − G (ghE + GEF P e
)
3
�
�
��
3
+2 e2B+W +U gX A hA − XA P A e−2B
2
�
��
�
��
1
U −W
A
A B C
+ e
(q̂A Az − CABC Az P Az ) − JeU −W
2
W +U +V
A
+2ge
hA P .
(6.20)
L = e
2B+W +U −V
This concludes the rewriting of the effective one-dimensional Lagrangian.
Setting the squares in (6.20) to zero yields the first-order flow equations
1
�
(AA
= − GAC q̂C eV −2B ,
z)
2
2
(X A )� =
X C (ghC eV + GCD P D eV −2B )X A − GAC (ghC eV + GCD P D eV −2B ) ,
3
U � − W � = Jˆ e−2B−2W +V ,
1
3
0 = B � − (U � + W � ) + XA P A eV −2B ,
2�
� 2
1
3
0 = 3 B � + (U � + W � ) − 2gX A hA eV + XA P A eV −2B .
(6.21)
2
2
These flow equations are supplemented by (6.11) and (6.16), and solutions to these equations are subjected to the constraint
hA P A = 0 ,
(6.22)
which follows from the last line of (6.20). Note that the flow equations (6.21) show an
interesting decoupling: The scalar fields X A and the metric coefficient eB are completely
determined by the magnetic fields and the fluxes, whereas the electric charges only enter
in the equations for the metric functions eU and eW and the AA
z -components of the gauge
fields. This will be helpful in the search for solutions, cf. sec. 6.3.
108
Subtracting the fourth from the fifth equation in (6.21) gives
B � + U � + W � = g hA X A e V .
(6.23)
When B is constant, this yields a flow equation for U + W that, when compared with the
fourth equation of (6.21), yields the condition
g X A hA = 3XA P A e−2B .
(6.24)
Also observe that (6.3), (6.11) and the first equation of (6.21) implies
1
FrtA = (1 − Ĉ) e−2B−W +U +V GAB q̂B ,
2
(6.25)
Ĉ ≡ C e−(U −W ) ,
(6.26)
where
while the third equation, together with (6.16), gives
and hence
�
��
C � = (U � − W � ) eU −W = eU −W ,
Ĉ = 1 + λ e−(U −W ) ,
(6.27)
(6.28)
with λ a real integration constant.1 Inserting this into (6.25) gives
λ
FrtA = − e−2B+V GAB q̂B .
2
(6.29)
However, the electric field is actually given by
1
A tz rr
(F A )tr = FtrA g tt g rr + Fzr
g g = e−2B−W −U −V GAB q̂B ,
2
(6.30)
where we used the form of the inverse metric, the third equation of (6.3), together with
the first equation of (6.21), and (6.29). Comparing this with (6.11), we see that this can
also be expressed as
�
(F A )tr = e−2U −2V ((eA )� + AA
zC ) .
(6.31)
Obviously, the electric field is independent of the integration constant λ and is nonvanishing whenever some of the charges q̂A are non-vanishing.
In contrast, the five-dimensional magnetic field component (F A )rz does depend on λ
according to
1
A rr zz
(F A )rz = Frz
g g + FrtA g rr g tz = λe−2B−W −U −V GAB q̂B .
2
(6.32)
Given the relation C = eU −W + λ between three of the metric functions, the solution set of the firstorder equations (6.21) is naturally more restricted than the one obtained by looking at the second-order
equations of motion. In particular, the charged magnetic brane solution of [138, 141] is not a solution of
(6.21).
1
109
Notice, however, that both the electric field and the rz-component of the magnetic field
A
BM N
are determined by the charges q̂A . As a consequence, the combination GAB FM
NF
vanishes (independently of λ) for vanishing P A on any solution of (6.21), i.e.
A Brz
GAB FrtA F Brt = −GAB Frz
F
.
(6.33)
On the other hand, inserting (6.28) into the line element (6.1) results in
�
�
ds2 = e2W λ λ + 2 eU −W dt2 + 2e2W C dt dz + e2W dz 2 + e2V dr2 + e2B (dx2 + dy 2 ) . (6.34)
Thus, we see that the sign and the magnitude of the integration constant λ determine the
nature of the warped line element. In particular, a vanishing λ will give a null-warped
metric, i.e. gtt = 0.
Let us now briefly display the flow equations for static, purely magnetic solutions. They
�
ˆ
are obtained by setting QA = AA
z = J = 0, which results in q̂A = J = C = 0, so that the
non-vanishing flow equations are
2 C
X (ghC eV + GCD P D eV −2B )X A − GAC (ghC eV + GCD P D eV −2B ) ,
3
U� − W � = 0 ,
1
3
0 = B � − (U � + W � ) + XA P A eV −2B ,
2�
� 2
1 �
3
�
�
0 = 3 B + (U + W ) − 2gX A hA eV + XA P A eV −2B .
(6.35)
2
2
(X A )� =
These flow equations need again to be supplemented by the constraint hA P A = 0. Magnetic
supersymmetric AdS3 × R2 solutions to these equations were studied in [136, 139, 140, 10].
Finally, we would like to show that the flow equations (6.21) follow from a superpotential. To do so, it is convenient to introduce the scalars
1
φ1 = B − (U + W ) ,
2
1
φ1 = B + (U + W ) ,
2
φ3 = U − W .
(6.36)
Using them and introducing the physical scalars ϕi , the one-dimensional Lagrangian (6.18)
takes the form
1
L = −e2φ2 −V (φ�1 )2 + 3e2φ2 −V (φ�2 )2 − e2φ2 −V (φ�3 )2
2
�
B �
−e2φ2 −V Gij (ϕi )� (ϕj )� − eφ1 +φ2 +φ3 −V GAB (AA
z ) (Az )
1
−e−2φ1 +V GAB P A P B − e−φ1 −φ2 +φ3 +V GAB q̂A q̂B
4
1 −2φ2 +2φ3 +V ˆ2
− e
J − g 2 e2φ2 +V (GAB hA hB − 2(hA X A )2 ) ,
2
(6.37)
where we used (cf. section 2.4 with gij replaced by Gij )
Gij = GAB ∂i X A ∂j X B .
(6.38)
110
The advantage of working with the scalars (6.36) is that the sigma-model metric is then
block diagonal with
gφ1 φ1 = e2φ2 −V
gij = e2φ2 −V Gij
1
gφ3 φ3 = e2φ2 −V ,
2
= eφ1 +φ2 +φ3 −V GAB .
gφ2 φ2 = −3e2φ2 −V
,
,
gAB
,
(6.39)
It is now a straightforward exercise to show that the potential V of the one-dimensional
Lagrangian (i.e. the last two lines of (6.37)) can be expressed as
�
�2
�
�2
�
�2
∂Z
∂Z
∂Z
∂Z ∂Z
∂Z ∂Z
φ1 φ1
φ2 φ2
φ3 φ3
V = −g
−g
−g
− g ij i j − g AB A
,(6.40)
∂φ1
∂φ2
∂φ3
∂ϕ ∂ϕ
∂Az ∂AB
z
with the superpotential
1
3
Z = eφ3 Jˆ + eφ2 −φ1 P A XA − e2φ2 ghA X A .
2
2
(6.41)
In doing so, one has to make use of the constraint hA P A = 0, of (6.38) and the identities
of section 2.4
2
Gij ∂i X A ∂j X B = GAB − X A X B
3
,
2
∂i XA = − GAB ∂i X B
3
,
2
XA = GAB X B .(6.42)
3
Using the superpotential (6.41), it is straightforward to check that the first-order flow
equations (in the physical moduli space) can be expressed as2
φ�1 = g φ1 φ1
∂Z
∂Z
∂Z
∂Z
�
AB ∂Z
, φ�2 = g φ2 φ2
, φ�3 = g φ3 φ3
, (AA
, (ϕi )� = g ij j ,
z) = g
B
∂φ1
∂φ2
∂φ3
∂Az
∂ϕ
(6.43)
In order to derive the flow equation for ϕi , one has to multiply the flow equation for X A
by GAB ∂j X B and use (6.38), (6.42) and
XA ∂ i X A = 0 .
6.1.2
(6.44)
Hamiltonian constraint
Next, we discuss the Hamiltonian constraint and show that it equals the constraint hA P A =
0 that we encountered in the rewriting of the Lagrangian in terms of first-order flow equations.
The Einstein equations take the form
RM N =
�
� 1
1
A
gM N g 2 GAB hA hB − 2(hA X A )2 − gM N GAB FKL
F B KL
3
6
A
B KL
+GAB (X A )� (X B )� δM r δN r + GAB FM
.
K FN L g
2
Note that (6.40) would hold also for any combination of signs in Z =
e2φ2 hA X A , but (6.43) requires the signs given in (6.41).
1 φ3 ˆ
2e J
(6.45)
± 32 eφ2 −φ1 P A XA ±
111
There are only five independent equations, namely the ones corresponding to the tt-, rr-,
xx-,zz- and tz-component of the Ricci tensor, which we have displayed in appendix C. To
obtain the Hamiltonian constraint, we consider the tt-component of Einstein’s equations.
We use the rr-, xx-,zz- and tz-equations to obtain expressions for the second derivatives
U �� , C �� , B �� and W �� , which we then insert into the expression for the tt-component. This
yields the following equation, which now only contains first derivatives,
1
0 = 2B �2 + 2U � W � + 4B � W � + 4B � U � + e2W −2U C �2 − GAB (X A )� (X B )� + GAB FrtA FrtB e−2U
2
�
B � −2W
� −2U
+GAB P A P B e2V −4B − GAB (AA
− e−2U C 2 ) − 2GAB FrtA (AB
C
z ) (Az ) (e
z )e
�
�
2 2V
AB
A 2
+g e
G hA hB − 2(hA X ) .
(6.46)
This equation turns out to be equivalent to the rr-component of Einstein’s equations,
1
1
δLM
Rrr − grr R − grr LM + rr = 0 ,
2
2
δg
(6.47)
where LM denotes the matter Lagrangian.
�
Next, using (6.11), (6.16), (6.27) and the flow equation for (AA
z ) in (6.46), we obtain
the intermediate result
1
0 = 2B �2 + 2U � W � + 4B � W � + 4B � U � + (U � − W � )2 − GAB (X A )� (X B )�
2
�
�
+GAB P A P B e2V −4B + g 2 e2V GAB hA hB − 2(hA X A )2 .
(6.48)
Then, using the first-order flow equation (6.21) for (X A )� , we get
1
4
0 = 2B �2 + 2U � W � + 4B � W � + 4B � U � + (U � − W � )2 − g 2 e2V (hA X A )2
2
3
2
4 2V −2B
A B 2 2V −4B
A
+ (GAB X P ) e
+ ge
(hA X )(GAB X A P B ) − 2ge2V −2B hA P A .
3
3
(6.49)
In the next step we use (6.23) as well as the fourth flow equation of (6.21) to obtain
1
4
0 = 2B �2 + 2U � W � + 4B � W � + 4B � U � + (U � − W � )2 − (B � + U � + W � )2
2
3
�
�
�2
�
2
1
4
1 �
�
�
�
�
�
�
�
�
+
−B + (U + W ) + (B + U + W ) −B + (U + W )
3
2
3
2
2V −2B
A
−2ge
hA P .
(6.50)
Then, one checks that all the terms containing B � , U � and W � cancel out, so that the
on-shell Hamiltonian constraint (6.50) reduces to (6.22).
112
6.2
Reducing to four dimensions
The five-dimensional stationary solutions to the flow equations (6.21) may be related to a
subset of the four-dimensional static solutions discussed in [119] and chapter 5 by performing a reduction on the z-direction. We briefly describe this below. A detailed check of the
matching of the five- and four-dimensional flow equations is performed in appendix D.
5d
The five-dimensional solutions are supported by electric fluxes h5d
A , electric charges QA ,
A
3
magnetic fields P5d , and rotation J. The relevant subset of four-dimensional solutions is
A
supported by electric fluxes h4d
A , electric charges QI = (Q0 , QA ) and magnetic fields P4d .
The five-dimensional N = 2, U (1) gauged supergravity action (6.4) is based on real
A
B
C
scalar fields X5d which satisfy the constraint 16 CABC X5d
X5d
X5D
= 1 for some constants
CABC , while the four-dimensional N = 2, U (1) gauged supergravity action considered in
I
[119] and chapter 5 is based on complex scalar fields X4D
with a cubic prepotential function
F (X4d ) = −
A
B
C
1 CABC X4d
X4d
X4d
.
0
6
X4d
(6.51)
A
0
The four-dimensional physical scalar fields are z A = X4d
/X4d
, which we decompose as
A
A
A
z = C + iX̂ .
Now we relate the real four-dimensional fields (C A , X̂ A ) to the fields appearing in
the five-dimensional flow equations. To do so, we find it convenient to use a different
normalization for the scalar constraint equation, namely
1
A
B
C
CABC X5d
X5d
X5D
=v .
6
(6.52)
We will show in the appendix that the matching between the four-dimensional and the fivedimensional flow equations requires to choose v = 12 , a value which was already obtained
in [64] when matching the gauge kinetic terms in four and five dimensions. Choosing the
normalization (6.52) amounts to replacing CABC by CABC /v, a change that affects the
normalization of the Chern-Simons term in the five-dimensional action (6.4), as well as the
quantities q̂A and Jˆ given in (6.12) and (6.17), respectively. On the other hand, if we stick
to the definition XA5d , we get for XA5d and GAB the expressions (2.81) and (2.79) which we
repeat here for convenience:
XA5d =
2v
B
GAB X5d
,
3
(6.53)
with GAB given by
1
GAB (X5d ) =
v
�
1
9
C
− CABC X5d
+ XA5d XB5d
2
2v
�
.
(6.54)
3
J generates translations in the z-direction. When the coordinate z is compact, J has the interpretation
of angular momentum.
6.3 Solutions
113
Using this normalization, we obtain the following dictionary between the four-dimensional
quantities that appeared in chapter 5 and the five-dimensional quantities that enter in
(6.21),
X̂ A
CA
h4d
A
A
P4d
Q4d
A
Q4d
0
A
eW X5d
,
A
Az ,
−h5d
A ,
A
−P5d
,
1 5d
= − QA ,
2
1
=
J.
2
=
=
=
=
(6.55)
The five- and four-dimensional line elements are related by
ds25 = e2φ ds24 + e−4φ (dz + Cdt)2 ,
(6.56)
where
ds24 = −e2U4 dt2 + e−2U4 dr2 + e−2U4 +2ψ (dx2 + dy 2 )
(6.57)
2φ = −W .
(6.58)
and
This yields
U = U4 + φ ,
V = −U4 + φ ,
B = ψ − U4 + φ .
6.3
(6.59)
Solutions
In the following, we construct solutions to the flow equations (6.21). First we consider exact
solutions with constant scalars X A . Subsequently we numerically construct solutions with
running scalars X A .
6.3.1
Solutions with constant scalar fields X A
We pick V = 0 in the following. We will consider two distinct cases. In the first case, all
the magnetic fields P A are taken to be non-vanishing. In the second case, we set all the
P A to zero. Other cases where only some of the P A are turned on are also possible, and
their analysis should go along similar lines.
114
Taking P A �= 0, q̂A = 0, Jˆ �= 0
Here we consider the case when all the P A are turned on. Demanding X A = constant
yields
X C (ghC + GCD P D e−2B )XA = ghA + GAB P B e−2B .
(6.60)
Observe that GAB is constant, and so is B. We set B = 0 in the following, which can
always be achieved by rescaling x and y. Combining (6.60) with (6.24), we express the
magnetic fields P A in terms of hA and X A as
�
�
�
3�
A
AB
C
P = −g G
hB −
hC X XB .
(6.61)
2
This relation generically fixes the scalars X A = X A (hB , P B ) in terms of the fluxes and
magnetic fields, as we will see in the explicit examples of sec. 6.3.2. Contracting (6.61)
with hA and using the constraint (6.22) we obtain
as well as
�
�2
GAB hA hB = hA X A
�2
1 �
GAB P A P B = g 2 hA X A .
2
(6.62)
(6.63)
Observe that (6.62) together with hA X A = 0 would imply hA = 0. Thus, in the following,
we take hA X A �= 0.
We obtain from (6.23),
U + W = ghA X A (r − r0 ) ,
(6.64)
where r0 denotes an integration constant. Inserting this into the third equation of (6.21)
gives
� −(U −W ) ��
A
e
= −Jˆ eghA X (r0 −r) .
(6.65)
Next we set q̂A = 0, so that the AA
z take constant values. These are determined by
QA + CABC P B AC
z = 0 .
(6.66)
Defining CAB = CABC P C , this is solved by
AB
AA
QB ,
z = −C
(6.67)
where C AB CBC = δ A C . Here, we assumed that CAB is invertible, which generically is the
case when all the P A are turned on.
A
ˆ
For constant AA
z , J is also constant, and we can solve (6.65). Taking hA X �= 0, we get
e−(U −W ) =
A
Jˆ eghA X (r0 −r)
+b,
ghA X A
(6.68)
6.3 Solutions
115
where b denotes an integration constant. Combining this result with (6.64) gives
e2W =
Jˆ
A
+ b eghA X (r−r0 )
A
ghA X
(6.69)
as well as
A
Jˆ e2ghA X (r0 −r)
A
e
=
+ b eghA X (r0 −r) .
(6.70)
A
ghA X
To bring these expressions into a more palatable form, we introduce a new radial variable
−2U
τ = α eghA X
A
(r−r0 )
with τ ≥ 0 and
α = ghA X A > 0 .
(If ghA X A < 0 we have τ ≤ 0.) Then (assuming Jˆ ≥ 0)
�
�
e2W = α−1 Jˆ + b τ ,
�
�
Jˆ
b
−2U
e
= α 2+
,
τ
τ
τ
C =
+λ,
ˆ
J + bτ
(6.71)
(6.72)
(6.73)
and the associated line element reads
α−2
dt2 + 2 dτ 2
τ
Jˆ + b τ
�
� �2
�
��
τ
−1
ˆ
+α
J + bτ
dz +
+ λ dt
Jˆ + b τ
� 2
�
+ dx + dy 2 ,
ds2 = −α−1
Now we notice that for
τ2
(6.74)
1
and b = 4α−3 > 0
(6.75)
b
and assuming z to be compact, this is nothing but the metric of the extremal BTZ black
hole in AdS3 times R2 , so that the space time is asymptotically AdS3 × R2 . This can be
made manifest by the coordinate redefinitions
λ=−
τ = ρ2 −
Jˆ
,
b
z=
l
φ,
b
(6.76)
where l2 = α b = 4/(ghA X A )2 . Introducing
j=
2Jˆ
,
bl
(6.77)
116
the line element becomes
�
�2
�
�−2
�
�2
�
�
ρ
j
ρ
j
j
ds2 = −
−
dt2 +
−
dρ2 + ρ2 dφ − 2 dt + dx2 + dy 2 .(6.78)
l
2ρ
l
2ρ
2ρ
This describes an extremal BTZ black hole with angular momentum j and mass M = j/l
ˆ
[145], where l denotes the radius of AdS3 . The horizon is at ρ2+ = jl/2 = J/b,
which
corresponds to τ = 0. The entropy of the BTZ black hole (and hence the entropy density
of the extremal BTZ ×R2 solution (6.78)) is
�
2π ρ+
Jˆ 3/2
SBTZ =
=π
α .
(6.79)
4
4
Observe that α is determined in terms of the fluxes hA and the P A through (6.61), and so
it is independent of J and QA .
In deriving the above solution, we have assumed that all the P A are turned on so as to
ensure the invertibility of the matrix CAB . In this generic case, the constant values of the
A
scalar fields X A and AA
and QA . When
z are entirely determined in terms of the hA , P
A
A
switching off some of the P , some of the Az may be allowed to have arbitrary constant
values, but these are expected not to contribute to the entropy density.
The BTZ×R2 -solution given above can be found in any N = 2, U (1) gauged super�
gravity model. This can also be inferred as follows. Setting B � = (X A )� = (AA
z ) = q̂A = 0
in the Lagrangian (6.18) and using the relations (6.62) and (6.63) yields a one-dimensional
Lagrangian that descends from a three-dimensional Lagrangian describing Einstein gravity in the presence of an anti-de Sitter cosmological constant Λ = −1/l2 determined by
the flux potential (4/l2 = (ghA X A )2 ). As is well-known, the associated three-dimensional
equations of motion allow for extremal BTZ black hole solutions with rotation. As shown
in [136], the near-horizon geometry of the BTZ×R2 -solution (which is supported by the
magnetic fields (6.61)) preserves half of the supersymmetry.
ˆ which takes the form
The entropy of the BTZ black hole solution depends on J,
J+
1 AB
C QA QB .
2
(6.80)
This combination is invariant under the transformation
J → Jˆ ,
QA → q̂A ,
(6.81)
with Jˆ and q̂A given in (6.17) and (6.12), respectively. In the absence of fluxes, this
transformation is called spectral flow transformation and can be understood as follows
from the supergravity perspective [11]. The rewriting of the five-dimensional Lagrangian
ˆ These
in terms of first-order flow equations makes use of the combinations q̂A and J.
combinations have their origin in the presence of the gauge Chern-Simons term. When the
ˆ
AA
z are constant, the shifts QA → q̂A and J → J take the form of shifts induced by a large
A
A
A
gauge transformation of Az , i.e. A → A + k A , where k A denotes a closed one-form.
6.3 Solutions
117
These transformations constitute a symmetry of string theory, and this implies that the
entropy of a black hole should be invariant under spectral flow. It must therefore depend
on the combination (6.80). In the presence of fluxes, we find that the BTZ ×R2 -solution
(6.78) respects the spectral flow transformation (6.81).
The three-dimensional extremal BTZ black hole geometry, resulting from dimensionally
reducing the black brane solution (6.78) on R2 , is a state in the two-dimensional CFT dual
ρ2+
3l
c
to the asymptotic AdS3 , with left-moving central charge c = 2G
and
L
−
=
, as
0
24
4 G3 l
3
c̃
well as L̃0 − 24 = 0, cf. [146]. Hence, the large charge leading term in the entropy of the
black hole is given by the Ramanujan-Hardy-Cardy formula for the dual CFT,
� �
c
c�
SBTZ = 2 π
L0 −
.
(6.82)
6
24
This is exactly equal to the Bekenstein-Hawking entropy (6.79) computed above (in units
of G3 = 1), and can be regarded as a microscopic computation of the bulk black brane
entropy from the holographic dual CFT.
Taking P A = 0, QA = 0, Jˆ �= 0
Now we consider the case when all the P A vanish. Then, (6.60) reduces to
�
�
hC X C XA = hA ,
(6.83)
which determines the constants X A in terms of the fluxes hA . Contracting (6.83) with
GAB hB yields
�2
2�
GAB hA hB =
hA X A .
(6.84)
3
Thus we take hA X A �= 0 in the following, since otherwise hA = 0.
Combining the fourth equation of (6.21) with (6.23) results in
1
B � = ghA X A ,
3
(6.85)
which can be readily integrated to give
1
eB = eβ e 3 ghA X
A
r
,
(6.86)
where β denotes an integration constant which we set to zero. The combination U + W is
given by
U + W = 2B + u ,
(6.87)
where u denotes an integration constant which we also set to zero.
The flow equation for U − W reads
�
e−(U −W )
��
4
A
= −Jˆ e−4B = −Jˆ e− 3 ghA X r .
(6.88)
118
Next, let us consider the flow equation for AA
z,
�
AA
z
��
= − 12 GAC QC e−2B .
(6.89)
2
A
AB
A non-vanishing QA yields a running scalar field AA
QB e− 3 ghA X r . In the chosen
z ∼ G
coordinates, the line element can only have a throat at |r| = ∞. At either of these points,
B
A
either the area element eB or AA
z blows up. If we demand that both e and Az stay finite
A
at the horizon, we are thus led to take Az to be constant, which can be obtained by setting
QA = 0. Therefore, we set QA = 0 in the following. This implies that Jˆ is constant, which
we take to be non-vanishing.
Taking hA X A �= 0, (6.88) is solved by
e−(U −W ) =
4
3 Jˆ
A
e− 3 ghA X r + γ ,
A
4 ghA X
(6.90)
where γ denotes an integration constant. Using (6.87), this results in
2
2
3 Jˆ
− ghA X A r
gh X A r
3
3 A
e
+
γ
e
,
A
4 ghA X
2
3 Jˆ
− ghA X A r
−2ghA X A r
3
=
e
+
γ
e
.
4 ghA X A
e2W =
e−2U
(6.91)
Redefining the radial coordinate,
1
τ = e 3 ghA X
A
r
,
τ ≥0,
(6.92)
yields
eB = τ ,
3 Jˆ
e2W =
τ −2 + γ τ 2 ,
4 ghA X A
3 Jˆ
e−2U =
τ −6 + γ τ −2 ,
4 ghA X A
τ4
C =
+λ.
3
Jˆ
+ γ τ4
4 ghA X A
(6.93)
In the chosen coordinates, the line element reads
2
ds = −e
2U (τ )
dt
2
+ ( 13 ghA X A )−2
�
dτ
τ
�2
+ e2B(τ ) (dx2 + dy 2 ) + e2W (τ ) (dz + C(τ )dt)2 . (6.94)
A
ˆ
It exhibits a throat as τ → 0. In the following we set λ = 0, and we take J/(gh
A X ) > 0.
6.3 Solutions
119
In the throat region, the terms proportional to γ do not contribute (recall that we are
taking Jˆ to be non-vanishing) and the line element becomes
� �2
dτ
2
6 2
A −2
1
ds = −τ dt + ( 3 ghA X )
+ τ 2 (dx2 + dy 2 ) + τ −2 (dz + τ 4 dt)2 ,
(6.95)
τ
where we rescaled the coordinates by various constant factors. Then, performing the
coordinate transformation
τ̃ = τ 3 , t̃ = 3t ,
(6.96)
and setting ( 13 ghA X A )2 = 1 for convenience, the line element becomes
�
� �2 �
�
�2
1
dτ̃
ds2 =
−τ̃ 2 dt̃2 +
+ τ̃ 2/3 (dx2 + dy 2 ) + τ̃ −2/3 dz + 13 τ̃ 4/3 dt̃
9
τ̃
� �2
1 dτ̃
2 2/3
−2/3
2
=
τ̃ dt̃dz + τ̃
dz +
+ τ̃ 2/3 (dx2 + dy 2 ) ,
3
9 τ̃
(6.97)
which describes a null-warped throat. The entropy density vanishes, S ∼ e2B+W |τ =0 = 0.
In the limit τ → ∞, on the other hand, there are two distinct cases. When γ �= 0 (we
take γ > 0),
eB =
e2W ≈
e−2U ≈
C ≈
τ,
γ τ2 ,
γ τ −2 ,
γ −1 ,
(6.98)
and the line element becomes
2
2
2
ds = −τ dt +
( 13 ghA X A )−2
�
dτ
τ
�2
+ τ 2 (dx2 + dy 2 ) + τ 2 (dz + dt)2 ,
(6.99)
where we rescaled the coordinates. Observe that this describes a patch of AdS5 .
The other case corresponds to setting γ = 0, in which case the behavior at τ → ∞ is
determined by
eB = τ ,
3
Jˆ
e2W =
τ −2 ,
4 ghA X A
3
Jˆ
e−2U =
τ −6 ,
4 ghA X A
4 ghA X A 4
C =
τ ,
3
Jˆ
and the associated line element is again of the form (6.95) and (6.97).
(6.100)
120
Thus, we conclude that the solution (6.94) with γ �= 0 describes a solution that interpolates between AdS5 and a null-warped Nernst throat at the horizon with vanishing
entropy, in which all the scalar fields are kept constant. This is a purely gravitational stationary solution that is supported by electric fluxes hA . It is an example of a Nernst brane
(i.e. a solution with vanishing entropy density), and is the five-dimensional counterpart of
the four-dimensional Nernst solution constructed in chapter 5. As already mentioned in
section 5.3 in Nernst geometries the tidal forces diverge.
6.3.2
Solutions with non-constant scalar fields X A
Here we present numerical solutions that are supported by non-constant scalar fields X A
and that interpolate between a near horizon solution of the type discussed above in sec.
6.3.1 and an asymptotic AdS5 -region with metric
ds2 = −e2r dt2 + dr2 + e2r (dx2 + dy 2 + dz 2 ) ,
(6.101)
i.e. the metric functions U (r), B(r) and W (r) in (6.1) all asymptote to the linear function
r and C becomes 0 (or constant, since a constant C can be removed by a redefinition of
the z-variable). To be concrete, we work within the STU-model. Within this model, a
solution with a single running scalar and with Jˆ = 0 was already given in sec. 2.3 of [10].
In order to facilitate the comparison with their results, we will work with physical scalars
in this section, i.e. we solve the constraint X 1 X 2 X 3 = 1 via
X1 = e
− √1 φ1 − √1 φ2
6
2
,
X2 = e
− √1 φ1 + √1 φ2
6
2
,
X3 = e
√2 φ1
6
.
(6.102)
In an asymptotically AdS5 -spacetime the two scalars φi have a leading order expansion
φi (r) = ai re−2r + bi e−2r + O(e−4r ) ,
i = 1, 2 ,
(6.103)
i.e. they both correspond to dimension 2 operators of the dual field theory and ai and bi
correspond to the sources and 1-point functions, respectively.
For all the following numerical solutions, we choose
g = 1 , h1 = h 2 = h 3 = 1 .
(6.104)
Solution with a single running scalar
ˆ For
Let us first consider the case with a single running scalar field and with vanishing J.
concreteness, we choose the magnetic fields
P 1 = P 2 = 41/3 ,
P 3 = −2 · 41/3 ,
(6.105)
so as to satisfy the constraint (6.22). One could choose a different overall normalization
for the magnetic fields P A by rescaling the x and y coordinates. This, on the other hand,
6.3 Solutions
121
would also imply a rescaling of eB and, thus, we would not have B = 0 anymore, as was
assumed in sec. 6.3.1. Solving (6.61) for these values of P A leads to
1
2
X(0)
= X(0)
= 4−1/3 ,
3
X(0)
= 42/3 ,
(6.106)
where we added the subscript (0) in order to distinguish the constant near horizon values
from the full, non-constant solution to be discussed momentarily. These values lead to
φ1(0) ≈ 1.132 ,
φ2(0) = 0 ,
α = 3 · 21/3 ≈ 3.780 ,
(6.107)
with α defined in (6.72). Finally, using (6.71), (6.73) and (6.75), we obtain
� α
� α
1 �
1 �
U(0) = − ln 4α−3 + r , W(0) = ln 4α−3 + r , C(0) = 0.
(6.108)
2
2
2
2
In order to obtain an interpolating solution with an asymptotic AdS5 -region, we slightly
perturb around this solution. In particuar, we make the following ansatz for small r
1
2
X 1 = X(0)
+ c1 ec2 r , X 2 = X(0)
+ c 3 e c4 r ,
B = δB , U = U(0) + δU , W = W(0) + δW
(6.109)
with c2 , c4 > 0. Plugging this into the flow equations (6.21), we obtain the following
conditions:
�√
�
c1 = c3 , c2 = c4 = 2−2/3
33 − 1 ≈ 2.989 ,
1 c3 (22/3 c2 + 14) c2 r
e , δB = 2−2/3 c3 (3 · 21/3 + c2 )ec2 r . (6.110)
2
2
The constant c3 is undetermined and sets the value of the source and 1-point function of
φ1 , cf. (6.103). We will see this explicitly in the example in sec. 6.3.2 below. Using (6.110),
we can find the initial conditions needed to solve (6.21) numerically. In practice it is most
convenient to solve (6.21) in the τ variable (6.71), as the horizon is at τ = 0. This allows
to set the initial conditions for instance at τ = 10−13 and then integrate outwards. Doing
so and choosing c3 = 1, we obtain the result depicted in figure 6.1 (note that the plot
makes use of the r-variable, i.e. the primes denote derivatives with respect to r as before).
Moreover, C ≡ 0. Even though the functions U and W always have the same derivative,
they differ by a shift, as can be seen in the right part of figure 6.1. This is due to the fact
that we chose b = 4α−3 according to (6.75), instead of b = 1, with b being introduced in
(6.68). This solution is very similar to the one discussed in sec. 2.3 of [10] and it is obvious
that the metric asymptotically becomes of the form (6.101).
δU = δW = −
Solution with two running scalars, Jˆ = 0
We now turn to the case of two non-trivial scalars, first still with vanishing Jˆ and then
with Jˆ �= 0 in the next subsection. In all cases we choose
� �1/3
� �1/3
� �1/3
4
4
4
1
2
3
P =
, P =2·
, P = −3 ·
.
(6.111)
3
3
3
122
U�
W�
r
φ1
U
B�
r
W
ˆ
Figure 6.1: The case of one scalar with vanishing J.
Again, the overall normalization of the P A is imposed on us by demanding B = 0 in the
near-horizon region. This time solving (6.61) leads to
� �2/3
� �2/3
1 4
4
3
1
2
3
X(0) =
, X(0) =
, X(0)
= 61/3 ,
(6.112)
4 3
3
2
which implies
φ1(0) ≈ 1.228 ,
7 2/3 1/3
· 4 · 3 ≈ 4.240 .
(6.113)
6
are again given by (6.108), now using the value of α given
φ2(0) ≈ 0.980 ,
α=
The functions U(0) , W(0) and C(0)
in (6.113).
Perturbing around the near-horizon solution utilizes the same ansatz as in (6.109). This
time, we obtain the conditions
c3 41 · 61/3 + 33c2
, c2 = c4 ≈ 3.694 ,
8 29 · 61/3 − 3c2
1 c3 (21 · 62/3 c22 + 588c2 − 178 · 61/3 ) c2 r
e ,
δU = δW =
16
c2 (−3c2 + 29 · 61/3 )
61/3 c3 (−27c22 − 42 · 61/3 c2 + 49 · 62/3 ) c2 r
δB =
e .
(6.114)
16
−3c2 + 29 · 61/3
Again, the parameter c3 determines the sources and 1-point functions of the scalars φ1 and
φ2 .
Using (6.114) in (6.109) we obtain the initial conditions to solve the flow equations
numerically. We do not find any solution for c3 = 1, but inverting the sign, i.e. choosing
c3 = −1, leads to the solution depicted in figure 6.2, which in addition has C ≡ 0 and
which is asymptotically AdS5 .
c1 = −
6.3 Solutions
123
U�
W�
φ1
φ2
B�
r
ˆ
Figure 6.2: The case of two scalars with vanishing J.
Solutions with two running scalars, Jˆ =
� 0
Finally, let us look at the more general case, where we have two running scalars and a
ˆ We choose the same values for the magnetic fields as in the last
constant non-vanishing J.
example, i.e. (6.111). Given that (6.61) does not depend on Jˆ at all, it is not surprising
A
that this leads to the same values for the X(0)
as in (6.112) (and, thus, also (6.113) does
not change). The main change arises for U, W and C, as their flow equations explicitly
ˆ They take on the near-horizon form
depend on J.
�
�
�
�
1
Jˆ 4eαr
1
Jˆ 4eαr
U(0) = − ln
+ 3
+ αr , W(0) = ln
+ 3
,
2
α
α
2
α
α
�
�
α3
1
C(0) =
−1 .
(6.115)
4 Jˆα42 e−αr + 1
Notice, in particular, the different behavior of U(0) and W(0) very close to the horizon, i.e.
ˆ
for r → −∞. Whereas the slope of U(0) and W(0) was α/2 in the case of vanishing J,
ˆ We will
cf. (6.108), now it is α for U(0) and zero for W(0) in the case of non-vanishing J.
clearly see this in the numerical solutions.
Again, we perturb around the near-horizon solution by (6.109). We again infer that
c2 = c4 and that c1 and c3 are related as in (6.114). Moreover, δU, δW and δB are all
proportional to ec2 r . Without going into the details, we present the resulting numerical
solutions for different values of Jˆ in figure 6.3. All these plots were produced using c3 =
−0.1 and b = 4α−3 . One can nicely see that the main difference to the case of vanishing
Jˆ appears in the U and W sector. The different slope of U and W close to the horizon,
ˆ U and W
mentioned in the last paragraph, is apparent. It is also obvious that for small J,
124
U�
4
U�
Jˆ = 10−9 b
4
3
3
2
φ1
φ2
2
φ1
φ2
1
W�
�6
B�
�4
�2
2
U�
4
6
1
W
r
�6
�4
�
B
2
W
1
B
�
B�
r
2
φ1
φ2
1
�2
6
3
2
�4
4
Jˆ = 15b
4
3
�6
�
�2
U�
Jˆ = b
4
φ1
φ2
Jˆ = 10−3 b
2
4
6
r
�6
�4
�2
�
W�
2
4
6
r
ˆ
Figure 6.3: The case of two scalars witht non-vanishing J.
first behave as in the case with vanishing Jˆ when approaching the horizon from infinity. I.e.
ˆ
they start out showing the same slope of α/2 until the J-term
starts dominating very close
to the horizon, where the slope of U doubles and W becomes constant. With increasing Jˆ
the intermediate region, where U and W have the same slope of α/2, becomes smaller and
smaller and finally disappears altogether.
The function C = eU −W − 1/b is shown (for Jˆ = b) in figure 6.4. Obviously, asymptotically it becomes constant and, thus, the asymptotic region is indeed given by AdS5 .
Finally, in figure 6.5, we plot φ1 and φ2 , multiplied with e2r , for two different values
of c3 . As expected from (6.103), the graphs show a linear behavior with non-vanishing
sources and 1-point functions for the two operators dual to the scalars. Obviously, these
sources and 1-point functions depend on the value of c3 .
6.3 Solutions
125
�6
�4
�2
2
4
r
6
�5
�10
�15
C
Figure 6.4: The function C for Jˆ = b.
150
150
c3 = −0.1
c3 = −0.01
100
φ1 e2r
50
�2
φ1 e2r
100
2
4
6
r 2r
φ2 e
50
�2
2
Figure 6.5: The two scalars multiplied with e2r .
4
6
r
φ2 e2r
126
Chapter 7
Conclusions and Outlook
In the last chapters we have met some interesting results within the wide subject of the
AdS/CFT correspondence.
We have seen how to apply AdS/CFT techniques to calculate the frequency dependent
conductivity tensor for field theories dual to a black hole in Einstein-Yang-Mills theory with
SU (2) gauge group. Further, we have constructed several new black solutions in N = 2
U (1) gauged supergravity in four and five dimensions. The larger part of these solutions
behave asymptotically like AdS which makes them interesting within the AdS/CFT context. In addition we found extremal black branes with zero entropy density – the Nernst
branes.
Nonetheless we are left with some yet unsolved problems. It will be very interesting to see
what causes the negative entropy production rate we found in chapter 4 for the normal
state of the field theory. As already mentioned at the end of chapter 4, the next task
will be to see whether we can find an instability on the gravity side looking at the full
Einstein-Yang-Mills equations.
Also our work on supergravity solutions in four and five dimension exhibits some “loose
ends”. Since all our four-dimensional Nernst solutions were axion-free it would be nice to
find one with axions excited. Moreover, it would be interesting to see whether the singular
solutions with flowing γ could be cured by taking into account higher derivative corrections
or whether there exist non-singular solutions with non-constant γ.
In five dimensions we met problems when adding electric charge. At present we could
not find a dyonic solution and we had the impression that having electric charges and
having magnetic fields seemed to be somehow complementary to each other. We saw these
difficulties even at the beginning when we performed the first-order rewriting since the
first-order rewriting in chapter 6 leads to flow equations for the scalars X A which only
contain magnetic fields and fluxes but no electric charges. The latter only influence the
equations of motion for the X A in an indirect way. However, as we have seen in appendix
E it is possible to find different rewritings. It would be interesting to see if there is another
rewriting that contains electric charges directly in the flow equations for the X A or if
otherwise there is an explanation why we could not find a rewriting of that type. This
128
7. Conclusions and Outlook
would also lead to a deeper understanding and this is what one is actually searching for.
We can only learn something new when getting involved with the puzzles and problems
that come across. What would science be without open questions – or to say it with
Einstein’s words from the beginning, without the mysterious?
Appendix A
Special geometry
We already stated in section 2.3 that the Lagrangian describing the couplings of N = 2
vector multiplets to N = 2 supergravity is encoded in a holomorphic function F (X), called
the prepotential, that depends on n + 1 complex scalar fields X I (I = 0, . . . , n). Here,
n counts the number of physical scalar fields. We will repeat here some of the identities
displayed in section 2.3. Moreover, we will provide the foundations for our calculations in
chapter 5.
For the prepotential F (X) it is known that the coupling to supergravity requires it to be homogeneous of degree two, i.e. F (λX) = λ2 F (X), from which one derives the homogeneity
properties
FI = FIJ X J ,
FIJK X K = 0 ,
(A.1)
where FI = ∂F (X)/∂X I , FIJ = ∂ 2 F/∂X I ∂X J , etc. The X I are coordinates on the big
moduli space, while the physical scalar fields z i = X i /X 0 (i = 1, . . . , n) parametrize an
n-dimensional complex hypersurface, which is defined by the condition that the symplectic
vector (X I , FI (X)) satisfies the constraint
�
�
i X̄ I FI − F̄I X I = 1 .
(A.2)
This can be written as
where
− NIJ X I X̄ J = 1 ,
�
�
NIJ = −i FIJ − F̄IJ .
(A.3)
(A.4)
The constraint (A.3) is solved by setting
X I = eK(z,z̄)/2 X I (z) ,
(A.5)
where K(z, z̄) is the Kähler potential,
e−K(z,z̄) = |X 0 (z)|2 [−NIJ Z I Z̄ J ]
(A.6)
130
A. Special geometry
with Z I (z) = (Z 0 , Z i ) = (1, z i ). Writing
� �2
F (X) = X 0 F(z) ,
(A.7)
which is possible in view of the homogeneity of F (X), we obtain
�
�
F0 = X 0 2F(z) − z i Fi ,
(A.8)
where Fi = ∂F/∂z i . In addition, we compute
F00 = 2F − 2z i Fi + z i z j Fij ,
F0j = Fj − z i Fij ,
Fij = Fij ,
to obtain
and hence
� �
� �
��
��
− NIJ Z I Z̄ J = i 2 F − F̄ − z i − z̄ i Fi + F̄i ,
� �
� �
��
��
e−K(z,z̄) = i |X 0 (z)|2 2 F − F̄ − z i − z̄ i Fi + F̄i .
(A.9)
(A.10)
(A.11)
I
As already mentioned in chapter 2 the X (z) are defined projectively, i.e. modulo multiplication by an arbitrary holomorphic function,
X I (z) → e−f (z) X I (z) .
(A.12)
This transformation induces the Kähler transformation
K → K + f + f¯
(A.13)
on the Kähler potential, while on the symplectic vector (X I , FI (X)) it acts as a phase
transformation, i.e.
1
¯
(X I , FI (X)) → e− 2 (f −f ) (X I , FI (X)) .
(A.14)
i
The resulting geometry for the space of physical scalar fields z is a special Kähler geometry,
with Kähler metric
∂ 2 K(z, z̄)
gi̄ =
(A.15)
∂z i ∂ z̄ j
based on a Kähler potential of the special form (A.11).
Let us relate the Kähler metric (A.15) to the metric (A.4) on the big moduli space.
Differentiating e−K yields
�
�
∂k e−K = −∂k K e−K = i |X 0 (z)|2 Fk − F̄k − (z i − z̄ i )Fik + ∂k ln X 0 (z) e−K ,
∂k ∂l̄ e−K = [−∂k ∂l̄ K + ∂k K ∂l̄ K] e−K = [−gkl̄ + ∂k K ∂l̄ K] e−K
�
� �
= i |X 0 (z)|2 Fkl − F̄kl − ∂k ln X 0 (z) ∂l̄ ln X̄ 0 (z̄)
�
−∂k ln X 0 (z) ∂l̄ K − ∂k K ∂l̄ ln X̄ 0 (z̄) e−K .
(A.16)
131
Using (A.9) we have
�
�
Nij = −i Fij − F̄ij ,
(A.17)
and hence we infer from (A.16) that
gi̄ = Nij |X 0 |2 +
1
|X 0 (z)|2
Di X 0 (z)D̄ X̄ 0 (z̄) ,
(A.18)
where
Di X 0 (z) = ∂i X 0 (z) + ∂i K X 0 (z)
(A.19)
�
denotes the covariant derivative of X 0 (z) under the transformation (A.12), i.e. Di e−f X 0 (z) =
e−f Di X 0 (z).
Next, let us consider the combination NIJ Dµ X I Dµ X̄ J , where the space-time covariant
derivative Dµ reads
Dµ X I = ∂µ X I + iAµ X I = ∂µ X I +
�
�
1�
∂i K ∂µ z i − ∂ı̄ K∂µ z̄ i X I ,
2
(A.20)
which is a covariant derivative for U (1) transformations (A.14). The combination
NIJ Dµ X I Dµ X̄ J is thus invariant under U (1)-transformations. Observe that
Dµ X 0 = eK/2 Di X (0) (z) ∂µ z i .
(A.21)
Using (A.3) we obtain
1
Dµ X 0 Dµ X̄ 0
|X 0 |2
X0
X̄ 0
+ 0 NiJ X̄ J ∂µ z i Dµ X̄ 0 + 0 NIj X I ∂µ z̄ j Dµ X 0 .
X
X̄
NIJ Dµ X I Dµ X̄ J = |X 0 |2 Nij ∂µ z i ∂ µ z̄ j −
Next, using
X 0 X̄ J NkJ =
1
X 0 (z)
Dk X 0 (z) ,
(A.22)
(A.23)
as well as (A.18) and (A.21) we establish
NIJ Dµ X I Dµ X̄ J = gi̄ ∂µ z i ∂ µ z̄ j ,
(A.24)
which relates the kinetic term for the physical fields z i to the kinetic term for the fields
X I on the big moduli space. Observe that both sides of (A.24) are invariant under Kähler
transformations (A.13).
Using the relation (A.24), we express the bosonic Lagrangian (describing the coupling
of n vector multiplets to N = 2 U (1) gauged supergravity [147, 148]) in terms of the fields
X I of big moduli space,
I
I
L = 12 R − NIJ Dµ X I Dµ X̄ J + 14 ImNIJ Fµν
F µνJ − 14 ReNIJ Fµν
F̃ µνJ − g 2 V (X, X̄) , (A.25)
132
A. Special geometry
where
NIJ = F̄IJ + i
NIK X K NJL X L
,
X M NM N X N
(A.26)
which satisfies the relation
NIJ X J = FI .
(A.27)
The flux potential reads
V = g i̄ Di W D̄̄ W̄ − 3|W |2 ,
W = hI FI − hI X I ,
(A.28)
where Di X I = ∂i X I + 12 ∂i K X I . Here, (hI , hI ) denote the magnetic/electric fluxes. AdS4
with cosmological constant Λ = −3g 2 corresponds to a constant W with |W | = 1. Switching off the flux potential corresponds to setting g = 0.
Using the identity (see (23) of [125])
N IJ = g i̄ Di X I D̄̄ X̄ J − X I X̄ J ,
the flux potential can be expressed as
�
��
��
�
V (X, X̄) = g i̄ Di X I D̄̄ X̄ J − 3X I X̄ J hK FKI − hI hK F̄KJ − hJ
�
��
��
�
= N IJ − 2X I X̄ J hK FKI − hI hK F̄KJ − hJ
¯ − 2 |W̃ |2 ,
= N IJ ∂ W̃ ∂ W̃
(A.29)
J¯
(A.30)
where in the last line W̃ is expressed in terms of U (1)-invariant fields X̃ I ,
�
�
W̃ = hI FI (X̃) − hI X̃ I = hI FIJ − hJ X̃ J .
(A.31)
I
Appendix B
First-order rewriting in minimal
gauged supergravity in D = 4
The N = 2 Lagrangian in minimal gauged supergravity is given in terms of the prepotential
F = − i (X 0 )2 . Only a single scalar field, X 0 is turned on, whose absolute value is set to
a constant, |X 0 | = 14 by the constraint (A.3), and the consequent bulk 1-D Lagrangian
density for the metric ansatz (2.1) is given by
L1D = e2 A + 2 U [ − A�2 − 2 A� U � +
1 −2 U − 4 A
e
( |Q̂0 |2 − 3 g 2 |ĥ0 |2 e4 A ) ] .
4
(B.1)
We now perform a first-order rewriting for this Lagrangian inspired by the corresponding
rewriting for black holes in minimal gauged supergravity, as done in [149]. Denoting
φ1 �=
�
1
1
A , φ2 = U , W = eU ( |Q̂0 |2 + g 2 |ĥ0 |2 e4 A + γ eA ) ,1 and the matrix, m =
,
1
0
we can write the above Lagrangian density as
�
L1D = − e2 A + 2 U [ φa� mab φb +
1 −4 U − 4 A ab
e
m Wa Wb ] .
4
(B.2)
Here, W1 = WA is the derivative of W w.r.t A and W2 = WU is the derivative of W w.r.t
U , given explicitly as
WA
=
WU
=
e2 U
( 4 g 2 |ĥ0 |2 e4 A + γ eA ) ,
2W
W.
(B.3)
The first-order rewriting is then simply,
L1D = − e2 A + 2 U [mab ( φa� −
1
1 − 2 U − 2 A ac
1
�
e
m Wc ) ( φb − e− 2 U − 2 A mbd Wd ) ] − φa� Wa .
2
2
(B.4)
Here γ is an arbitrary real number.
134
B. First-order rewriting in minimal gauged supergravity in D = 4
Using the chain-rule of derivatives on W = W ( φa ) , the last term in the above equation
becomes φa � Wa = W � . Hence the first-order rewritten Lagrangian can finally be written
as
1 − 2 U − 2 A ac
1
�
e
m Wc ) ( φb − e− 2 U − 2 A mbd Wd ) ] − W � .
2
2
(B.5)
The first-order equations for the metric functions can be written as
L1D = − e2 A + 2 U [mab ( φa� −
A�
=
U�
=
1 −2 U − 2 A
e
WU ,
2
1 −2 U − 2 A
e
( WA − WU ) .
2
(B.6)
(B.7)
The Hamiltonian density can be written as H = L1D + [ e2 ( A+ U ) ( 2 A� ) ] . On-shell, using
the first-order equations, this can be written as H = L1D + [ e2 ( A+ U ) ( W � ) ] . Substituting
for L1D from (B.6), we see that the on-shell Hamiltonian density becomes
H = − e2 A + 2 U [mab ( φa� −
1 − 2 U − 2 A ac
1
�
e
m Wc ) ( φb − e− 2 U − 2 A mbd Wd ) ] .
2
2
(B.8)
This is trivially zero on-shell as the perfect squares vanish due to the first-order equationss. Hence the Hamiltonian constraint from General Relativity is satisfied for field
configurations obeying these first-order equations. This completes the first-order rewriting
for the Lagrangian in minimally gauged supergravity. All smooth black brane solutions
to these first-order first-order equationss are necessarily non-supersymmetric. The first
non-supersymmetric black brane solutions in minimally gauged supergravity were written
down in [8] where it was also shown that the supersymmetric solutions are singular.
There is a different rewriting of this Lagrangian which gives an AdS2 × R2 background,
given below as
L1D
=
�
1
�
�
− [ ( ( eA ) )2 e2 U + 2 eA + U ( ( eA ) ) ( eU − √ )
v1
1
1
�
+ 2 ( ( eA ) ) eA + U √ − e− 2 A ( |Q̂0 |2 − 3 g 2 |ĥ0 |2 e4 A ) ] .
v1
4
(B.9)
The first-order first-order equationss following from the rewriting above are
( eA )
U
(e )
4A
�
�
=
=
0,
r
√ ,
v1
(B.10)
(B.11)
2
e
with v1 = 2 |Q̂
. The Hamiltonian density vanishes on-shell provided e4 A = 3 g|Q̂2 |0ĥ| |2 . The
2
0|
0
AdS2 × R2 background which is the near-horizon black brane geometry in the presence of
fluxes, is a solution to these first-order equationss.
Appendix C
Einstein equations in five dimensions
When evaluated on the solution ansatz (6.1), the independent Einstein equations take the
following form:
tt-component:
�
1 −2(U +V )
e
− e2(U +W ) C �2 − 2e2(U +W ) C(2B � C � − C � (U � + V � − 3W � ) + C �� )
2
+2e4U (2B � U � + U �2 − U � V � + U � W � + U �� )
�
�
�
2W 2 2W �2
2U
�
�
�
�
�
�
�2
��
−e C e C + 2e (2B W + U W − V W + W + W )
�
�
� 1
1 2 � AB
2U
2W 2
A 2
A
B KL
= (−e + e C ) g G hA hB − 2(hA X ) − GAB FKL F
+ GAB FrtA FrtB e−2V
3
6
(C.1)
rr-component:
1
−2B �2 + e2W −2U C �2 − U �2 + 2B � V � + U � V � + V � W � − W �2 − 2B �� − U �� − W ��
2
�
�
� 1
1 2 � AB
A �
B �
2V
A 2
A
B KL
= GAB (X ) (X ) + e
g G hA hB − 2(hA X ) − GAB FKL F
3
6
� A B
�
−2U
A �
B � −2W
−2U 2
� −2U
+GAB Frt Frt (−e ) + (Az ) (Az ) (e
− e C ) + 2FrtA (AB
C
z )e
(C.2)
xx-component:
�
�
−e2B−2V 2B �2 + B � U � − B � V � + B � W � + B �� =
�
�
� 1
1 2 � AB
2B
A 2
A
B KL
+ GAB P A P B e−2B (C.3)
e
3
6
136
C. Einstein equations in five dimensions
zz-component:
�
1 2W −2U −2V � 2W �2
e
−e C − 2e2U (2B � W � + U � W � − W � V � + W �2 + W �� )
2
�
�
� 1
1 2 � AB
A 2
A
B KL
�
B � −2V
2W
= e
+ GAB (AA
z ) (Az ) e
3
6
(C.4)
tz-component:
1 2W −2U −2V
e
2
�
− 2e2U (B � C � + 2CB � W � ) + e2U (C � U � + C � V � − 3C � W � − C �� ) − e2W CC �2
�
2U
�
�
�
�
�2
��
+2e C(U W − V W + W + W )
�
�
� 1
1 2 � AB
2W
A 2
A
B KL
� −2V
+ 2GAB FrtA (AB
= e C
z )e
3
6
(C.5)
Appendix D
Relating five- and four-dimensional
flow equations
We relate the four-dimensional flow equations for black branes (5.34) in big moduli space
to the five-dimensional flow equations (6.21). We set g = 1 throughout. For literature on
the 4D − 5D connection see e.g. [150, 151].
The four-dimensional N = 2, U (1) gauged supergravity theory is based on complex
scalar fields X I , I = 0, . . . , n encoded in the cubic prepotential (with A = 1, . . . , n)
F (X) = −
1 CABC X A X B X C
= (X 0 )2 F(z) ,
6
X0
(D.1)
where z A = X A /X 0 denote the physical scalar fields and
1
F(z) = − CABC z A z B z C .
6
(D.2)
Differentiating with respect to z A yields
1
FA = − CABC z B z C ,
2
FAB = −CABC z C ,
(D.3)
where FA = ∂F/∂z A , etc. The Kähler potential K(z, z̄) is determined in terms of F by
�
�
e−K = i 2(F − F̄) − (z A − z̄ A )(FA + F̄A )
i
=
CABC (z A − z̄ A )(z B − z̄ B )(z C − z̄ C ) .
(D.4)
6
The Kähler metric gAB̄ = ∂A ∂B̄ K(z, z̄) can be expressed as
�
�
gAB̄ = KA KB̄ − ieK FAB − F̄AB ,
(D.5)
where KA = ∂K/∂z A and
KA = −KĀ = −eK 2i CABC (z B − z̄ B )(z C − z̄ C ) .
(D.6)
138
D. Relating five- and four-dimensional flow equations
In the following we pick the gauge X 0 (z) = 1, X A (z) = z A (with X I (z) ≡ X I e−K/2 ), so
that the complex scalar fields X I and the z A are related by X 0 = eK/2 and X A = eK/2 z A .
Using the dictionary (6.55) that relates the quantities appearing in the four- and fiveA
dimensional flow equations, in particular z A − z̄ A = 2ieW X5d
, we obtain
e−K = 8v e3W
(D.7)
as well as
KĀ = −
3i −W 5d
e
XA ,
2v
1
gAB̄ = 4veK+W GAB = e−2W GAB ,
2
(D.8)
where GAB denotes the target space metric in five dimensions, cf. (6.54), and the XA5d were
defined in (6.53). The factor of v in (D.7) arises due to the normalization in (6.52). Using
these expressions, we establish
g AB̄ KB̄ = −(z A − z̄ A ) .
(D.9)
In the big moduli space, the four-dimensional flow equations were expressed in terms
of rescaled complex scalar fields Y I given by Y 0 = |Y 0 |eiα and Y A = Y 0 z A , where |Y 0 | =
eK/2+ψ−U4 . On a solution to the four-dimensional flow equations we can relate the phase
α to the phase γ that enters in the four-dimensional flow equations. We obtain α = −γ,
which we establish as follows. Writing e2iα = Y 0 /Ȳ 0 we get
�
�
0
�
i −2iα (Y 0 )�
i �
0 � Y
�
α =− e
− (Ȳ )
= − 0 e−2iα (Y 0 )� − (Ȳ 0 )� .
(D.10)
0
0
2
2
Ȳ
(Ȳ )
2Ȳ
The flow equation for Y 0 reads (cf. (5.34))
(Y 0 )� = eψ−U4 N 0J q̄J
�
�
¯
¯
= eψ−U4 +K g AB̄ KA q̄J¯ (∂B̄ + KB̄ ) X̄ J (z̄) − q̄J¯ X̄ J (z̄) ,
(D.11)
where we used
�
�
N IJ = eK g AB̄ (∂A + ∂A K)X I (z) (∂B̄ + ∂B̄ K)X̄ J (z̄) − X I (z)X̄ J (z̄) ,
cf. for instance [125]. In (D.11) the qI denote the four-dimensional quantities
�
�
qI = eU4 −2ψ+iγ Q̂I − ie2(ψ−U4 ) ĥI .
(D.12)
(D.13)
The quantities Q̂I and ĥI are combinations of the four-dimensional charges and fluxes given
by (cf. (5.27),(5.28))
Q̂I = QI − FIJ P J ,
ĥI = hI − FIJ hJ .
(D.14)
139
For later use, we also introduce the quantities
Z(Y ) = −Q̂I Y I ,
W (Y ) = −ĥI Y I .
(D.15)
Inserting the flow equation (D.11) in (D.10) yields
��
��
�
i
α� = − eiα eK/2 g AB̄ KA KB̄ − 1 e−2iα (q̄0 + q̄C̄ z̄ C̄ ) − (q0 + qC z C )
2
�
+g AB̄ (e−2iα KA q̄B̄ − qA KB̄ ) ,
(D.16)
where we used the relation
�
�
q̄J (∂B̄ + KB̄ ) X̄ J (z) = q̄B̄ + KB̄ q̄0 + q̄A z̄ A
(D.17)
as well as |Y 0 | = eK/2+ψ−U4 . Next, using that on a four-dimensional solution we have
¯
qI Y I = q̄I Ȳ I , we obtain
e−2iα (q̄0 + q̄C̄ z̄ C̄ ) = q0 + qC z C .
(D.18)
Inserting this in (D.16) results in
�
�
i
α� = − eiα eK/2 g AB̄ e−2iα KA q̄B̄ − qA KB̄ .
2
(D.19)
Using (D.9), we obtain
�
�
�
�
g AB̄ e−2iα KA q̄B̄ − qA KB̄ = (z A − z̄ A ) e−2iα q̄Ā + qA
= e−2iα z A q̄Ā − z̄ A qA − e−2iα z̄ A q̄Ā + qA z A
= e−2iα (q̄0 + z A q̄Ā ) − (q0 + z̄ A qA ) ,
where we used (D.18) in the last equality. Now we notice that
�
�
qI z̄ I = q0 + z̄ A qA = −ei(α+γ) e−K/2+2U4 −3ψ Z̄(Ȳ ) − ie2(ψ−U4 ) W̄ (Ȳ )
(D.20)
(D.21)
and
�
�
e−2iα q̄I¯z I = e−2iα (q̄0 + z A q̄Ā ) = −e−i(3α+γ) e−K/2+2U4 −3ψ Z(Y ) + ie2(ψ−U4 ) W (Y ) ,(D.22)
so that
�
�
�
�
α� = −e2U4 −3ψ Im e−i(2α+γ) Z(Y ) − e−ψ Re e−i(2α+γ) W (Y ) .
This is precisely the flow equation for γ, provided α = −γ.
(D.23)
140
Using this result, we now relate the flow equations for the z A to the five-dimensional
A
0
A
flow equations for X5d
and AA
z . Using the four-dimensional flow equations for Y and Y
we obtain
�
1 � A �
(z A )� =
(Y ) − z A (Y 0 )�
0
Y
�
eψ−U4 � AJ
=
N − z A N 0J q̄J
0
Y
�
�
−K
= e 2 +iγ N AJ − z A N 0J q̄J .
(D.24)
Then, using (D.12), one derives
N AJ − z A N 0J = eK g AB̄ (∂B̄ + KB̄ ) X̄ J (z) ,
which implies
(D.25)
K
(z A )� = e 2 +iγ g AB̄ q̄J (∂B̄ + KB̄ ) X̄ J (z) .
(D.26)
Now we specialize to four-dimensional solutions that are supported by electric charges QI ,
magnetic charges P A and electric fluxes hA . Decomposing z A = C A + iX̂ A and using the
expression (D.13) gives
�
1
1
A
U4 −2ψ−iγ
q̄0 + q̄A z̄ = e
Q0 + CABC P A C B C C + QA C A − CABC P A X̂ B X̂ C
2
2
�
��
+e2(ψ−U4 ) hA X̂ A + i −QA X̂ A − CABC P A C B X̂ C + e2(ψ−U4 ) hA C A
. (D.27)
Then, using (D.17) and (D.8) leads to (we recall (6.58))
�
�
K
1
A �
+U
−2ψ−4φ
AB
K−4φ
5d
4
(z ) = 2e 2
G
− 12ie
XB Q0 + CCDE P C C D C E
2
�
1
D
E −4φ
E −2φ
+ QE C E − CCDE P C X5d
X5d
e
+ e2(ψ−U4 ) hE X5d
e
2
+ QB + CBEF C E P F
+12eK−4φ XB5d
�
E −2φ
E −2φ
− QE X5d
e
− CCDE P C C D X5d
e
+ e2(ψ−U4 ) hE C E
�
� 2(ψ−U4 )
�
E F −2φ
+i e
hB − CBEF P X5d e
.
Thus, we obtain for the real part,
A �
(C )
= 2e
K
+U4 −2ψ−4φ
2
+12e
K−4φ
AB
G
�
XB5d
�
�
(D.28)
QB + CBEF C E P F
E
C
D
E
− QE X̂ − CCDE P C X̂ + e
2(ψ−U4 )
hE C
E
��
.
(D.29)
141
Next we show that the second line of this equation vanishes by virtue of the four-dimensional
flow constraint
�
�
�
�
Im eiγ Z(Y ) − e2(ψ−U4 ) Re eiγ W (Y ) = 0 .
(D.30)
We have
�
�
1
Z(Y ) = Y 0 − CABC P A z B z C − QA z A − Q0
2
(D.31)
W (Y ) = −Y 0 hA z A .
(D.32)
�
�
�
�
Im eiγ Z(Y ) = |Y 0 | −CABC P A C B X̂ C − QA X̂ A
(D.33)
and
This leads to
as well as
�
�
Re eiγ W (Y ) = −|Y 0 |hA C A .
This gives
(D.34)
�
�
|Y 0 | −CABC P A C B X̂ C − QA X̂ A + e2(ψ−U4 ) hA C A =
�
�
�
�
= Im eiγ Z(Y ) − e2(ψ−U4 ) Re eiγ W (Y ) ,
(D.35)
which vanishes due to (D.30), so that (D.29) becomes
�
�
1
(C A )� = √ eV −2B GAB QB + CBEF C E P F ,
2v
(D.36)
where we used the relations (6.59) and (D.7).
For the imaginary part of (z A )� we get
�
e
−2φ
�
A �
X5d
= 2e
K
+U4 −2ψ−4φ
2
�
−e
2φ
A
X5d
�
1
Q0 + CBCD P B C C C D + QB C B
2
�
1
B C
D −4φ
2(ψ−U4 )
B −2φ
− CBCD P X5d X5d e
+e
hB X5d e
2
�
� 2(ψ−U4 )
�
AB
E F −2φ
+G
e
hB − CBEF P X5d e
.
(D.37)
Using (6.54) we obtain
9
C
− CABC X5d
= 2vGAB − XA5d XB5d .
v
(D.38)
B
Contracting this expression once with P A X5d
and once with P A , we rewrite the two exB C
D
F
pressions containing CBCD P X5d X5d and CBEF P E X5d
in (D.37). Using (D.7) as well we
142
obtain
�
e
−2φ
�
A �
X5d
�
�
�
1 U4 −2ψ−φ
1
2φ A
B C D
B
= √ e
− e X5d Q0 + CBCD P C C + QB C
2
2v
=
�
A
+ 2vP A e−2φ − 3X5d
(XB5d P B )e−2φ
�
��
3 5d
C
2(ψ−U4 ) AB
+e
G
hB − XB (hC X5d )
2v
�� A
A �
e−2φ X5d
+ e−2φ (X5d
) .
A �
Using XA5d (X5d
) = 0 (and (6.58), (6.59)) we infer
�
�
�
� −2φ ��
1 U4 −2ψ−φ
1
2φ
B C D
B
e
= √ e
−e
Q0 + CBCD P C C + QB C
2
2v
�
1 2(ψ−U4 )
−2φ 5d B
C
− e XB P − e
hC X5d
3
�
�
�
1
1
−2B−2W +V −2φ
B C D
B
= −√
e
Q0 + CBCD P C C + QB C
2
2v
�
1 −U
V −2B−2φ 5d B
C
+e
XB P + e hC X5d
3
as well as
�
�
�
1
A �
A
(X5d
) = √ eU4 −2ψ+φ 2e−2φ vP A − X5d
(XB5d P B )
2v
�
��
1 5d
2(ψ−U4 ) AB
C
+e
G
hB − XB (hC X5d )
.
v
(D.39)
(D.40)
(D.41)
The former should match the flow equation for eW . To check this, we note that the fivedimensional flow equations (6.21) imply
2
1
A V
U � + W � = hA X5d
e + XA5d P A eV −2B .
3
v
(D.42)
Subtracting this from the third equation of (6.21) gives
1
1
1 ˆ −2B−2W +V
A V
W � = hA X5d
e + XA5d P A eV −2B − Je
,
3
2v
2
so that
� W ��
1
1
1 ˆ −2B−2W +V −2φ
A V −2φ
e
= e−2φ W � = hA X5d
e
+ XA5d P A eV −2B−2φ − Je
.
3
2v
2
(D.43)
(D.44)
This matches (D.40) if we set v = 1/2 and perform the identifications (6.55). Under these
A
identifications, the flow equations (D.36) and (D.41) precisely match those for AA
z and X5d
appearing in (6.21). Similarly, the flow equations for the four-dimensional warp factors U4
and ψ match those of the five-dimensional warp factors U and B using (6.59).
Appendix E
A different first-order rewriting in
five dimensions
We present a different first-order rewriting that allows for solutions with electric fields.
This rewriting is the one performed in [12] for static black hole solutions, which we adapt
to the case of stationary black branes in the presence of magnetic fields.
We consider the metric (6.1) and the gauge field ansatz (6.2) with AA
z = 0, so that
q̂A = QA and Jˆ = J. The starting point of the analysis is therefore Lagrangian (6.18),
ˆ
with AA
z = 0, q̂A = QA and J = J.
We perform the following g-split of U and V ,
1
log f ,
2
1
V = V0 − log f ,
2
f = f0 (r) + g 2 f2 (r) = −µ r2 + g 2 e2U2 (r) ,
V0 = 2B(r) + W (r) + U0 (r) + log(r) .
U = U0 +
(E.1)
In addition, we perform the rescaling
P A = g pA ,
J = gj ,
(E.2)
and we organize the terms in the Lagrangian into powers of g. This yields L = L0 + g 2 L2 .
First, we analyze L0 ,
�
L0 = e2B+W +U0 −V0 f0 2B �2 + 2U � W � + 4B � W � + 4B � U � − GAB (X A )� (X B )�
�
1 AB
−4B−2W +2V0
−1
− G QA QB e
(f0 )
.
(E.3)
4
We perform a first-order rewriting of L0 by introducing parameters q̃A and γA that are
related to the electric charges qA by
1 AB
G QA QB = −µ GAB q̃A γB .
4
(E.4)
144
E. A different first-order rewriting in five dimensions
We obtain
�
�
L0 = −µ r eU0 q̃A GAB 6XB + eU0 (r2 q̃B − γB )
�
1
2B+W +U0 −V0
+e
f0 (2B � + U0� )(2B � + 4W � + 3U0� )
2
�
�
2 −2B−W +V0 �
9 AB �
2 −2B−W +V0 ��
�
�
− G
XA − U0 XA + e
q̃A XB − U0 XB + e
q̃B
4
3
3
�
��
−2 eU0 q̃A X A f0 − µ(2W � + 4B � ) .
(E.5)
This yields the first-order flow equations
2
XA� = U0� XA − e−2B−W +V0 q̃A ,
3
1 �
�
B = − U0 ,
2
�
2B = −4W � − 3U0� .
(E.6)
Using (E.1) as well as X A (XA )� = 0, these equations yield
Integrating the latter gives
1
W � = B � = − U0�
2
� −U0 ��
2
A
e
= − r X q̃A ,
3
� −U0 ��
2
e XA
= − r q̃A .
3
e−U0 XA =
1
HA ,
3
HA = γ̃A − r2 q̃A ,
(E.7)
(E.8)
where γ̃A denote integration constants. Contracting this with X A results in
e−U0 =
1
HA X A ,
3
(E.9)
which satisfies the second equation of (E.7) by virtue of XA (X A )� = 0.
L0 contains, in addition, the first line, which is not the square (or the sum of squares)
of a first-order flow equation. Its variation with respect to U0 gives
�
�
��
q̃A GAB 3XB + eU0 r2 q̃B − γB = 0 .
(E.10)
Comparing with (E.8) yields
AB
q̃A GAB (γ̃B − γB ) = 0 .
(E.11)
Since G
is positive definite, we conclude that this can only be fulfilled for arbitrary
values of q̃A if γ̃B = γB . On the other hand, varying the first line of L0 with respect to X A
and using (E.8) gives
�
�
− 2µ r eU0 q̃A 2 δX A + GAC δGCD X D ,
(E.12)
145
which vanishes by virtue of δGCD X D = 3 δXC = −2 GCD δX D . Thus, we conclude that
the set of variational equations derived from L0 is consistent.
Now we turn to L2 ,
�
�
�2
1
L2 = e2B+W +U0 −V0 +2U2 −
j e−2B−2W +V0 −U2 + (W � − (U0� + U2� ))
2
�
�2
1
3
− B � − (U0� + U2� + W � ) + XA pA e−2B+V0 −U2
2
2
� �
�
�2
1
1 �
3
�
�
�
A
V0 −U2
A −2B+V0 −U2
+ 3 B + (U0 + U2 + W ) − 2X hA e
+ XA p e
3
2
2
�
�
��
2 C
�A
V0 −U2
D −2B+V0 −U2
A
AC
V0 −U2
D −2B+V0 −U2
−GAB X − X (hC e
+ GCD p e
)X − G (hC e
+ GCD p e
)
3
��
�
�
2
X �B − X E (hE eV0 −U2 + GEF pF e−2B+V0 −U2 )X B − GBE (hE eV0 −U2 + GEF pF e−2B+V0 −U2 )
3
�
�
��
3
2B+W +U0 +U2
A
A −2B
+2 e
X hA − XA p e
2
� −W +U0 +U2 ��
W +U0 +V0
− je
+ 2e
h A pA .
(E.13)
This yields the first-order flow equations
2 C
X (hC eV0 −U2 + GCD pD e−2B+V0 −U2 )X A − GAC (hC eV0 −U2 + GCD pD e−2B+V0 −U2 ) ,
3
W � = U0� + U2� − j e−2B−2W +V0 −U2 ,
1
3
0 = B � − (U0� + U2� + W � ) + XA pA e−2B+V0 −U2 ,
2
2
� � 1 �
�
3
0 = 3 B + 2 (U0 + U2� + W � ) − 2X A hA eV0 −U2 + XA pA e−2B+V0 −U2 ,
2
(E.14)
(X A )� =
as well as the constraint
h A pA = 0 .
(E.15)
We have thus derived two sets of first-order flow equations (one derived from L0 and the
other derived from L2 ) that need to be mutually consistent. Consistency of these two sets
implies certain relations which we now derive.
Adding the third and fourth equation of (E.14) gives
1
B � = X A hA eV0 −U2 − XA pA e−2B+V0 −U2 .
3
(E.16)
Combining this with the first equation of (E.7) yields
2
U0� = − X A hA eV0 −U2 + 2XA pA e−2B+V0 −U2 .
3
(E.17)
146
Comparing with the second equation of (E.7) yields
1 −2B−W A
1
e
X q̃A = − X A hA e−U2 + XA pA e−2B−U2 .
3
3
(E.18)
Contracting the first equation of (E.14) with GAB results in
2
2
XA� = −( X C hC eV0 −U2 + XC pC e−2B+V0 −U2 )XA + (hA eV0 −U2 + GAD pD e−2B+V0 −U2 ) ,
3
3
(E.19)
Using (E.17) gives
�
�
2
XA� = U0� − 3XC pC e−2B+V0 −U2 XA + (hA eV0 −U2 + GAD pD e−2B+V0 −U2 ) .
3
(E.20)
Using W � = − 12 U0� (from the first equation of (E.7)) in the second equation of (E.14)
gives
3
U2� = − U0� + j e−2B−2W +V0 −U2 ,
(E.21)
2
which expresses U2 in terms of U0 .
The second equation of (E.14) can be rewritten as
U0� + U2� + W � = 2W � + j e−2B−2W +V0 −U2
(E.22)
which, when inserted into the third equation of (E.14), gives
− j e−2W + 3XA pA = 0 .
(E.23)
Using −2W = U0 (ignoring an additive constant) as well as (E.8), this results in
j = HA p A .
(E.24)
Since the left hand side is constant, we conclude that
q̃A pA = 0 ,
j = γ A pA .
(E.25)
Finally, comparing the first equation of (E.6) with (E.20)
2
2
− 3XC pC e−2B−U2 XA + (hA e−U2 + GAD pD e−2B−U2 ) + q̃A e−2B−W = 0 .
3
3
(E.26)
Note that the contraction of (E.26) with X A gives back (E.18). On the other hand,
contracting (E.26) with pA and using (E.15) and (E.25) gives
�
�2 2
3 XA pA = pA GAB pB .
3
(E.27)
147
Observe that both sides are positive definite. This relation is, for instance, satisfied for the
STU-model X 1 X 2 X 3 = 1. Inserting the relation
1
9
GAB = − CABC X C + XA XB
2
2
(E.28)
X A CABC pB pC = 0 .
(E.29)
into (E.27) we obtain
Thus, we conclude that the two sets (E.7) and (E.14) are mutually consistent, provided
that (E.26) is satisfied and the constraints (E.15) and (E.25) hold.
In the following, we solve the first-order flow equations for the case when pA = 0. Then
j = 0, so that from (E.21) we obtain U2� = − 32 U0� , which also equals W � + 2B � by virtue of
the first equation of (E.7). Thus U2 = W + 2B, up to an additive constant. Then, (E.26)
is satisfied provided we set q̃A = −hA . Summarizing, when pA = 0, we obtain
1 AB
G QA QB = µ GAB hA γB ,
4
1
e−U0 XA =
HA , H A = γ A + r 2 h A ,
3
1
e−U0 =
HA X A ,
3
1
B = W = − U0 ,
2
3
U2 = − U0 ,
2
1
V0 = − U0 + log r ,
2
f = −µ r2 + g 2 e−3U0 ,
1 2U0
(eA )� =
e r GAB QB .
2
(E.30)
This is the black brane analog of the black hole solutions discussed in [12].
For this class of solutions, we now check the Hamiltonian constraint
Rtt +
δLM
1
− gtt (R + LM ) = 0 ,
δg tt
2
where LM denotes the matter Lagrangian. Using
�
�
�
�
�
��
√
1
−g g tt Rtt − gtt R = −3 e3B+U −V B �2 + B � U � + 3 e3B+U −V B � ,
2
(E.31)
(E.32)
as well as
�
�
�
�
δLM
1
1
− gtt LM = gtt e−6B GAB QA QB + e−2V GAB X �A X �B + g 2 GAB − 2X A X B hA hB ,
tt
δg
2
2
(E.33)
148
where we replaced the electric fields by their charges, we obtain for (E.31),
�
��
L0 + L2 − 6 e3B+U −V B � = 0 .
(E.34)
Imposing the first-order flow equations, this reduces to
� U0
��
�
��
�
��
e f0 q̃A X A + 3µ B + g 2 e3B+U0 +U2 q̃A X A + 3 e3B+U0 −V0 f B �
+µ r eU0 q̃A X A = 0 .
(E.35)
Then, using the second equation of (E.7) and
3 r B �� = −U0� eU0 r2 q̃A X A − eU0 r q̃A X A − eU0 r2 q̃A X �A ,
(E.36)
we find that (E.35) is satisfied on a solution to the first-order flow equations. Thus, the
Hamiltonian constraint (E.31) does not lead to any further constraint.
The class of static solutions (E.30) was obtained long time ago in [13] by solving the
equations of motion. Their mass density is determined by µ. Let us consider a black
solution, with the horizon located at f (rH ) = 0 with e2U0 (rH ) �= 0. Its temperature is then
given by
 3

� U0 −V0
�
U0
2
e
e
TH =
|f � |
=
|f � |
,
(E.37)
4π
4π r
r=rH
r=rH
where we used (E.30) to express V0 in terms of U0 . The horizon condition f (rH ) = 0 gives
2
µ rH
= g 2 e−3U0 (rH ) ,
(E.38)
and hence rH �= 0. For the solution to be extremal, its temperature has to vanish. Imposing
f � (rH ) = 0 results in
3
µ rH + g 2 e−3U0 (rH ) U0� (rH ) = 0 .
(E.39)
2
Combining both equations gives
1+
3
rH U0� (rH ) = 0 .
2
(E.40)
As an example, consider the STU-model X 1 X 2 X 3 = 1 and take γA = c hA with c > 0, so
that HA = hA H with H = c + r2 . Then
e−3U0 = H1 H2 H3 = h1 h2 h3 H 3
X2 =
(H1 H3 )1/3
2/3
H2
=
(h1 h3 )1/3
2/3
h2
,
,
X1 =
X3 =
(H2 H3 )1/3
2/3
H1
(H1 H2 )1/3
2/3
H3
=
=
(h2 h3 )1/3
,
2/3
h1
(h1 h2 )1/3
2/3
h3
.
(E.41)
The scalars X A are thus constant. We set g = 1 and α3 = h1 h2 h3 > 0. The horizon
condition (E.38) yields
µ H(rH ) − α3 H 3 (rH ) = µ c ,
(E.42)
149
while the condition (E.39) gives
�
�
rH µ − 3α3 H 2 (rH ) = 0 ,
(E.43)
which implies (we take rH �= 0),
µ = 3α3 H 2 (rH ) .
(E.44)
2α3 H 3 (rH ) = µ c .
(E.45)
Inserting (E.44) into (E.42) gives
Combining (E.45) with the first equation of (E.30), which takes the form
µc =
1 QA GAB QB
,
4 hA GAB hB
(E.46)
yields the entropy density as
�1/2 1 � µ c �1/2 1
1 3B(rH ) 1 � 3 3
S= e
=
α H (rH )
=
=
4
4
4 2
8
�
1 QA GAB QB
2 hA GAB hB
�1/2
.
Extremal electric solutions of this type have been considered recently in [14, 10].
(E.47)
150
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Danksagung
Zunächst möchte ich mich bei Michael Haack für die Zusammenarbeit während der letzten drei Jahre bedanken. Während dieser Zeit hatte ich die Möglichkeit an interessanten
Themen zu arbeiten. Besonders bedanken möchte ich mich auch für Michael Haacks Unterstützung. Er hatte stets Zeit für meine Fragen.
Desweiteren gilt mein Dank Dieter Lüst, der mir die Möglichkeit gab im Rahmen des
IMPRS-Graduiertenprogramms zu promovieren. An seinem Lehrstuhl herrscht eine freundliche Atmosphäre, die es ermöglicht frei und produktiv zu arbeiten.
Weiterhin möchte ich mit bei allen bedanken, mit denen ich im Rahmen der Promotion
außerdem zusammengearbeitet habe, nämlich Gabriel Lopes Cardoso, Suresh Nampuri,
Niels Obers und Giuseppe Policastro. Besonders erwähnen möchte ich an dieser Stelle
Suresh Nampuri und Gabriel Lopes Cardoso. Suresh Nampuri hatte stets kreative Ideen
und brachte immer Tempo in unsere Besprechungen (ich kenne niemanden, der schneller
sprechen und schreiben kann als er!). Bei Gabriel Lopes Cardoso möchte ich mich für die
Beharrlichkeit bedanken, mit der er unsere Zusammenarbeit vorantrieb und niemals locker
ließ, bevor ein Problem gelöst war.
Während meiner Zeit an der LMU konnte ich jedoch nicht nur im akademischen Bereich die Bekanntschaft hilfsbereiter Menschen machen. Ich denke, ich spreche wohl für
viele Studenten und Doktoranden, wenn ich mich an dieser Stelle bei Bernhard Emmer
bedanke. Er nimmt sich immer Zeit bei administrativen Problemen und sucht beharrlich
nach einer Lösung.
Zuletzt gilt mein Dank meinem Ehemann Andreas. Er war immer für mich da, wenn
ich Entscheidungen zu treffen hatte.

Aspects of AdS / CFT - Elektronische Dissertationen der LMU

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